Preface

Fracture mechanics deals with the cracking behavior of materials, and cracking defines the limit state for many components of engineering systems. Therefore, fracture mechanics is a field that provides useful tools for many engineering practitioners and designers from various disciplines. It is a relatively new branch of mechanics and more advanced in theoretical basis rather than practical applications. However, due to its useful approach in handling failure of materials, its use is expected to be more widespread in the near future.

Fracture mechanics principles can help us design more robust components ensuring safer airplanes, space shuttles, ships, cranes, buildings, bridges and other mechanical systems for use in medicine, for example, destroying arterial plaques or kidney stones. In short, application of fracture mechanics has potential to give us tools in solution of many design and analysis problems. In this context, this book introduces and supports the understanding of fracture mechanics principles to enhance the knowledge and application of specialists in all related areas of engineering and science.

*Fracture Mechanics Applications* contains eight chapters written by researchers and experts in the field. It is designed for graduate students, researchers, and practicing engineers.

Chapter 1 introduces rupture theory of soft-matter quasicrystals. Chapters 2 provides insight into application of the J-integral for the fracture assessment of welded polymeric components. Chapter 3 describes fatigue limit reliability analysis for notched material with some kinds of dense inhomogeneities using fracture mechanics. Chapter 4 deals with the fracture behavior of pure and hybrid intraply knitted fabric-reinforced polymer composites. Chapter 5 explains the construction of an artificial neural network to detect structural damage. Chapter 6 introduces the mechanism of misalignment of saw cutting cracks in concrete pavement. Chapter 7 presents the effect of geometrical factors on the surface pressure distribution on a human phantom model following shock exposure. Finally, Chapter 8 discusses blast effects on different structural elements.

We would like to express our deepest appreciation to academicians, scientists, and our colleagues and friends globally who have significantly contributed in the broad field of fracture mechanics, and especially to the authors of the chapters in this book. We are also grateful to IntechOpen for publishing the book, which we hope to be a noteworthy contribution to the current literature.

**Chapter 1**

**Abstract**

**1. Introduction**

tals is very important.

**1**

Probe on Rupture Theory of

In this chapter, a probe on an important aspect, rupture theory of soft matter, is discussed, in which the soft matter and soft-matter quasicrystals are introduced at first. Then, we discuss the behaviour of the matter. For the soft-matter quasicrystals, there are basic equations describing their dynamics; due to the work of the second author of this chapter, this provides a fundamental for studying the rupture feature quantitatively. For general soft matter, there are no such equations so far, whose rupture theory quantitatively is not available at moment. The discussion on the softmatter quasicrystals may provide a reference for other branches of soft matter.

**Keywords:** soft matter, quasicrystals, generalized dynamics, equation of state,

Soft-matter quasicrystals belong to a category of soft matter. It is well known that the soft matter is an intermediate phase between solid and liquid, which presents the behaviour of both solid and liquid, the first nature of which is fluidity, and behaves as a complex structure. Hence, the soft matter is named as a complex liquid or a structured liquid. Soft-matter quasicrystals are soft-matter with quasi-periodic symmetry. In this sense, they are a category of soft matter with highest symmetry so far. The high symmetry of the matter presents symmetry breaking and leads to importance of elementary excitations. This helps us to set up their dynamic equa-

The liquid crystals, polymers, colloids, surfactants, and so on are in common named as soft matter. The 12-fold symmetry quasicrystals were observed most frequently in the soft matter [1–7]. At the same time, cracks in soft matter [8, 9] cannot be ignored and should be prevented [10] from the material safety. Other crack and rupture problems in soft matter can be referred from [11–18]. This shows that the study on crack and rupture for soft matter including soft-matter quasicrys-

It is well known that the failure of brittle structural materials has been well studied. According to the work of Griffith [19], the existence and propagation of crack is the reason of the failure of these materials. Griffith used the exact solution of a crack in an infinite plate and calculated the crack strain energy. The energy is

crack, stress intensity factor, generalized Griffith criterion of rupture

tions and provide the fundamental for the rupture study.

**2. Soft matter and the rupture problem**

Soft-Matter Quasicrystals

*Hui Cheng and Tian-You Fan*

## **Chapter 1**

## Probe on Rupture Theory of Soft-Matter Quasicrystals

*Hui Cheng and Tian-You Fan*

## **Abstract**

In this chapter, a probe on an important aspect, rupture theory of soft matter, is discussed, in which the soft matter and soft-matter quasicrystals are introduced at first. Then, we discuss the behaviour of the matter. For the soft-matter quasicrystals, there are basic equations describing their dynamics; due to the work of the second author of this chapter, this provides a fundamental for studying the rupture feature quantitatively. For general soft matter, there are no such equations so far, whose rupture theory quantitatively is not available at moment. The discussion on the softmatter quasicrystals may provide a reference for other branches of soft matter.

**Keywords:** soft matter, quasicrystals, generalized dynamics, equation of state, crack, stress intensity factor, generalized Griffith criterion of rupture

## **1. Introduction**

Soft-matter quasicrystals belong to a category of soft matter. It is well known that the soft matter is an intermediate phase between solid and liquid, which presents the behaviour of both solid and liquid, the first nature of which is fluidity, and behaves as a complex structure. Hence, the soft matter is named as a complex liquid or a structured liquid. Soft-matter quasicrystals are soft-matter with quasi-periodic symmetry. In this sense, they are a category of soft matter with highest symmetry so far. The high symmetry of the matter presents symmetry breaking and leads to importance of elementary excitations. This helps us to set up their dynamic equations and provide the fundamental for the rupture study.

## **2. Soft matter and the rupture problem**

The liquid crystals, polymers, colloids, surfactants, and so on are in common named as soft matter. The 12-fold symmetry quasicrystals were observed most frequently in the soft matter [1–7]. At the same time, cracks in soft matter [8, 9] cannot be ignored and should be prevented [10] from the material safety. Other crack and rupture problems in soft matter can be referred from [11–18]. This shows that the study on crack and rupture for soft matter including soft-matter quasicrystals is very important.

It is well known that the failure of brittle structural materials has been well studied. According to the work of Griffith [19], the existence and propagation of crack is the reason of the failure of these materials. Griffith used the exact solution of a crack in an infinite plate and calculated the crack strain energy. The energy is

the function of crack size. He further calculated crack energy release rate and suggested that when the release rate equals to the surface tensile of the material, the solid will be a failure. His hypothesis was proved by experiments. This is the famous Griffith criterion. Afterward, the classical Griffith theory was developed by Irwin et al. for studying quasi-brittle failure, where the strain energy release rate was replaced by so-called stress intensity factor and the surface tensile was replaced by the fracture toughness of the material. This is the engineering approach of the Griffith theory, and played an important role in engineering application. The failure of ductile materials is also related with the existence and propagation of crack, but the problem has not been well studied due to the plastic deformation around the crack tip. The plastic deformation is a nonlinear irreducible process. The problem is extremely complex physically and mathematically. The failure of soft matter will be more complex than the ductile structural materials because of the existence of the fluidity. Especially, the experimental results are few of reported. As a most preliminary probe for studying the problem of soft matter, we try to draw from the idea of the classical Griffith work, i.e., to study the crack stability/instability, we can use the so-called following Griffith-Irwin criterion

$$K\_1(a, \sigma) = K\_{IC}(T\_\bullet f(s)) \tag{1}$$

the equation of state as well, which is also completed by Fan and co-worker [28]. With

For the need of the present chapter, we list the two-dimensional equations of the

*<sup>∂</sup><sup>x</sup>* <sup>þ</sup> *<sup>η</sup>*∇<sup>2</sup>

*Vx* þ 1 3 *η ∂ <sup>∂</sup><sup>x</sup>* <sup>∇</sup> � **<sup>V</sup>**

*Vy* þ 1 3 *η ∂ ∂y* ∇ � **V**

> � *∂*2 *wx ∂y*<sup>2</sup>

*wy ∂y*<sup>2</sup>

> *∂y* ∇ � **u**þ

*wx ∂y*<sup>2</sup>

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>;

*<sup>∂</sup><sup>x</sup>* <sup>∇</sup> � **<sup>u</sup>**<sup>þ</sup>

*∂*2 *wy ∂x∂y*

� �

*ux* <sup>þ</sup> ð Þ *<sup>L</sup>* <sup>þ</sup> *<sup>M</sup> <sup>∂</sup>*

*∂*2 *wx <sup>∂</sup>x∂<sup>y</sup>* � *<sup>∂</sup>*<sup>2</sup>

� �#

*uy* <sup>þ</sup> ð Þ *<sup>L</sup>* <sup>þ</sup> *<sup>M</sup> <sup>∂</sup>*

*∂*2 *wy <sup>∂</sup>x∂<sup>y</sup>* � *<sup>∂</sup>*<sup>2</sup>

� �#

*ux ∂x∂y*

� �#

*∂*2 *uy ∂x∂y*

� �#

� *∂*2 *uy ∂y*<sup>2</sup>

� *∂*2 *ux ∂y*<sup>2</sup>

*wy ∂y*<sup>2</sup>

9

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>=

(2)

*∂*2 *wx <sup>∂</sup>x∂<sup>y</sup>* � *<sup>∂</sup>*<sup>2</sup>

� �

these bases, the generalized dynamics of soft-matter quasicrystals is set up.

*∂ VyρVx* � �

*<sup>∂</sup><sup>x</sup>* <sup>∇</sup> � **<sup>u</sup>**

*∂ VyρVy* � �

> *∂y* ∇ � **u**

*wy ∂y*<sup>2</sup>

*∂ρ ∂t*

*<sup>∂</sup>*ð Þ *<sup>ρ</sup>Vx ∂t* þ

<sup>þ</sup>*M*∇<sup>2</sup>

*∂*2 *wx <sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> <sup>2</sup>

�ð Þ <sup>A</sup> � <sup>B</sup> <sup>1</sup>

*ρ*0 *∂δρ ∂x*

*∂ VxρVy* � � *∂x* þ

*uy* <sup>þ</sup> ð Þ *<sup>L</sup>* <sup>þ</sup> *<sup>M</sup>* � *<sup>B</sup> <sup>∂</sup>*

*∂*2 *wx <sup>∂</sup>x∂<sup>y</sup>* � *<sup>∂</sup>*<sup>2</sup>

� �

*∂ux*

*∂uy*

*wy ∂y*<sup>2</sup>

*∂wx*

� *∂*2 *ux ∂y*<sup>2</sup>

� *∂*2 *uy ∂y*<sup>2</sup>

� �

*∂wy*

þ

*∂*2 *wy <sup>∂</sup>x*<sup>2</sup> � <sup>2</sup>

�ð Þ <sup>A</sup> � <sup>B</sup> <sup>1</sup>

*ρ*0 *∂δρ ∂y*

*∂ux <sup>∂</sup><sup>x</sup>* <sup>þ</sup> *Vy*

*∂uy <sup>∂</sup><sup>x</sup>* <sup>þ</sup> *Vy*

*∂*2 *wy <sup>∂</sup>x∂<sup>y</sup>* � *<sup>∂</sup>wx ∂y*<sup>2</sup>

*∂*2 *wx <sup>∂</sup>x∂<sup>y</sup>* � *<sup>∂</sup>*<sup>2</sup>

*∂*2 *uy ∂x∂y*

� �

*ux ∂x∂y*

� �

*ρ ρ*0 þ *ρ*2 *ρ*0 <sup>2</sup> <sup>þ</sup> *<sup>ρ</sup>*<sup>3</sup> *ρ*0 3

*∂wx <sup>∂</sup><sup>x</sup>* <sup>þ</sup> *Vy*

*∂wy <sup>∂</sup><sup>x</sup>* <sup>þ</sup> *Vy*

� �

� �

þ*R*<sup>1</sup>

*∂ ρVy* � � *∂t*

<sup>þ</sup>*M*∇<sup>2</sup>

þ*R*<sup>1</sup>

*∂ux ∂t* þ *Vx*

*R*1 *∂*2 *wx <sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> <sup>2</sup>

*∂uy ∂t* þ *Vx*

*R*1 *∂*2 *wy <sup>∂</sup>x*<sup>2</sup> � <sup>2</sup>

*∂wx ∂t*

*R*1 *∂*2 *ux <sup>∂</sup>x*<sup>2</sup> � <sup>2</sup>

*∂wy ∂t*

*R*1 *∂*2 *uy <sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> <sup>2</sup> *<sup>∂</sup>*<sup>2</sup>

*p* ¼ 3

**3**

þ *Vx*

þ *Vx*

*kBT l* 3

þ ∇ � ð Þ¼ *ρ***V** 0

*<sup>∂</sup>*ð Þ *VxρVx ∂x* þ

*Probe on Rupture Theory of Soft-Matter Quasicrystals DOI: http://dx.doi.org/10.5772/intechopen.89813*

*ux* <sup>þ</sup> ð Þ *<sup>L</sup>* <sup>þ</sup> *<sup>M</sup>* � *<sup>B</sup> <sup>∂</sup>*

*∂*2 *wy ∂x∂y*

� �

� *∂*2 *wx ∂y*<sup>2</sup>

dynamics of soft-matter quasicrystals of 5- and 10-fold symmetry as follows:

*<sup>∂</sup><sup>y</sup>* ¼ � *<sup>∂</sup><sup>p</sup>*

� *R*<sup>2</sup>

*<sup>∂</sup><sup>y</sup>* ¼ � *<sup>∂</sup><sup>p</sup>*

þ *R*<sup>2</sup>

*<sup>∂</sup><sup>y</sup>* <sup>¼</sup> *Vx* <sup>þ</sup> <sup>Γ</sup>**u**½*M*∇<sup>2</sup>

� *R*<sup>2</sup>

*<sup>∂</sup><sup>y</sup>* <sup>¼</sup> *Vy* <sup>þ</sup> <sup>Γ</sup>**u**½*M*∇<sup>2</sup>

þ *R*<sup>2</sup>

*<sup>∂</sup><sup>y</sup>* <sup>¼</sup> <sup>Γ</sup>**w**½*K*1∇<sup>2</sup>

þ *R*<sup>2</sup>

*<sup>∂</sup><sup>y</sup>* <sup>¼</sup> <sup>Γ</sup>**w**½*K*1∇<sup>2</sup>

� *R*<sup>2</sup>

and *Rijkl* and *Rklij* are the phonon-phason coupling elastic constant tensors.

*∂*2 *wy <sup>∂</sup>x*<sup>2</sup> � <sup>2</sup>

*∂*2 *wx <sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> <sup>2</sup>

*∂*2 *uy <sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> <sup>2</sup> *<sup>∂</sup>*<sup>2</sup>

*∂*2 *ux <sup>∂</sup>x*<sup>2</sup> � <sup>2</sup>

*wx*þ

*wy*þ

in which *ui* denotes the phonon field, *wi* the phason field, *Vi* the fluid velocity field, *Cijkl* is the phonon elastic constant tensor, *Kijkl* phason elastic constant tensor,

*∂*2 *wy <sup>∂</sup>x*<sup>2</sup> � <sup>2</sup>

> *∂y* <sup>þ</sup> *<sup>η</sup>*∇<sup>2</sup>

*∂*2 *wx <sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> <sup>2</sup>

in which *K*1ð Þ *a*, *σ* represents elastic stress intensity factor, which a function of crack size *a* and applied external stress *σ*, can be determined by stress analysis of cracked materials, and the *KIC*ð Þ *T*, *f s*ð Þ fracture toughness, a material constant but influenced by temperature and the structure factor *f s*ð Þ, and the suffice I expresses mode I, i.e., the opening mode fracture (and the mode II is shearing mode, or slip mode fracture, and mode III is longitudinal shear mode, or tearing mode fracture, we here discuss only the opening mode). If the value of *K*1ð Þ *a*, *σ* is greater than that of *KIC*ð Þ *T*, *f s*ð Þ , then the crack will propagate and the material will fracture.

Of course, the criterion (1) is only a reference for the soft matter, and a further analysis will be given in the subsequent sections.

## **3. Soft-matter quasicrystals and their generalized dynamics**

There are a quite lot of references concerning the crack and rupture problems in soft matter; however, the quantitative analyses are not so much, because most branches of soft matter science are in qualitative stage so far. Either theoretical research or engineering application, the rupture problem of the soft matter needs a quantitative analysis.

Recently, generalized dynamics of soft-matter quasicrystals has been developed [20–24], which may become another quantitative branch in soft matter apart from the liquid crystals science. The generalized dynamics of soft-matter quasicrystals provides a tool for analyzing quantitatively rupture problem of the matter, whose result may be references of other categories of soft matter.

Soft-matter quasicrystals look like other categories of soft matter which belong to complex fluid; at mean time they present highly symmetry. For the high ordered phase, the symmetry breaking and elementary excitation principle are important. By using the Landau-Anderson [25, 26] symmetry breaking and elementary excitation principle, there are phonon and phason elementary excitations. As a class of soft matter, the fluidity is the substantive nature of the soft-matter quasicrystals; so Fan [20–24] introduced another elementary excitation—fluid phonon apart from phonons and phasons; of course, the concept of the fluid phonon is originated from the Landau School [27]. The introducing of fluid phonon requires a supplemented equation and

the function of crack size. He further calculated crack energy release rate and suggested that when the release rate equals to the surface tensile of the material, the solid will be a failure. His hypothesis was proved by experiments. This is the famous Griffith criterion. Afterward, the classical Griffith theory was developed by Irwin et al. for studying quasi-brittle failure, where the strain energy release rate was replaced by so-called stress intensity factor and the surface tensile was replaced by the fracture toughness of the material. This is the engineering approach of the Griffith theory, and played an important role in engineering application. The failure of ductile materials is also related with the existence and propagation of crack, but the problem has not been well studied due to the plastic deformation around the crack tip. The plastic deformation is a nonlinear irreducible process. The problem is extremely complex physically and mathematically. The failure of soft matter will be more complex than the ductile structural materials because of the existence of the fluidity. Especially, the experimental results are few of reported. As a most preliminary probe for studying the problem of soft matter, we try to draw from the idea of the classical Griffith work, i.e., to study the crack stability/instability, we can use

in which *K*1ð Þ *a*, *σ* represents elastic stress intensity factor, which a function of crack size *a* and applied external stress *σ*, can be determined by stress analysis of cracked materials, and the *KIC*ð Þ *T*, *f s*ð Þ fracture toughness, a material constant but influenced by temperature and the structure factor *f s*ð Þ, and the suffice I expresses mode I, i.e., the opening mode fracture (and the mode II is shearing mode, or slip mode fracture, and mode III is longitudinal shear mode, or tearing mode fracture, we here discuss only the opening mode). If the value of *K*1ð Þ *a*, *σ* is greater than that

Of course, the criterion (1) is only a reference for the soft matter, and a further

There are a quite lot of references concerning the crack and rupture problems in

Recently, generalized dynamics of soft-matter quasicrystals has been developed [20–24], which may become another quantitative branch in soft matter apart from the liquid crystals science. The generalized dynamics of soft-matter quasicrystals provides a tool for analyzing quantitatively rupture problem of the matter, whose

Soft-matter quasicrystals look like other categories of soft matter which belong to complex fluid; at mean time they present highly symmetry. For the high ordered phase, the symmetry breaking and elementary excitation principle are important. By using the Landau-Anderson [25, 26] symmetry breaking and elementary excitation principle, there are phonon and phason elementary excitations. As a class of soft matter, the fluidity is the substantive nature of the soft-matter quasicrystals; so Fan [20–24] introduced another elementary excitation—fluid phonon apart from phonons and phasons; of course, the concept of the fluid phonon is originated from the Landau School [27]. The introducing of fluid phonon requires a supplemented equation and

soft matter; however, the quantitative analyses are not so much, because most branches of soft matter science are in qualitative stage so far. Either theoretical research or engineering application, the rupture problem of the soft matter needs a

of *KIC*ð Þ *T*, *f s*ð Þ , then the crack will propagate and the material will fracture.

**3. Soft-matter quasicrystals and their generalized dynamics**

result may be references of other categories of soft matter.

*K*1ð Þ¼ *a*, *σ KIC*ð Þ *T*, *f s*ð Þ (1)

the so-called following Griffith-Irwin criterion

*Fracture Mechanics Applications*

analysis will be given in the subsequent sections.

quantitative analysis.

**2**

the equation of state as well, which is also completed by Fan and co-worker [28]. With these bases, the generalized dynamics of soft-matter quasicrystals is set up.

For the need of the present chapter, we list the two-dimensional equations of the dynamics of soft-matter quasicrystals of 5- and 10-fold symmetry as follows:

*∂ρ ∂t* þ ∇ � ð Þ¼ *ρ***V** 0 *<sup>∂</sup>*ð Þ *<sup>ρ</sup>Vx ∂t* þ *<sup>∂</sup>*ð Þ *VxρVx ∂x* þ *∂ VyρVx* � � *<sup>∂</sup><sup>y</sup>* ¼ � *<sup>∂</sup><sup>p</sup> <sup>∂</sup><sup>x</sup>* <sup>þ</sup> *<sup>η</sup>*∇<sup>2</sup> *Vx* þ 1 3 *η ∂ <sup>∂</sup><sup>x</sup>* <sup>∇</sup> � **<sup>V</sup>** <sup>þ</sup>*M*∇<sup>2</sup> *ux* <sup>þ</sup> ð Þ *<sup>L</sup>* <sup>þ</sup> *<sup>M</sup>* � *<sup>B</sup> <sup>∂</sup> <sup>∂</sup><sup>x</sup>* <sup>∇</sup> � **<sup>u</sup>** þ*R*<sup>1</sup> *∂*2 *wx <sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> <sup>2</sup> *∂*2 *wy ∂x∂y* � *∂*2 *wx ∂y*<sup>2</sup> � � � *R*<sup>2</sup> *∂*2 *wy <sup>∂</sup>x*<sup>2</sup> � <sup>2</sup> *∂*2 *wx <sup>∂</sup>x∂<sup>y</sup>* � *<sup>∂</sup>*<sup>2</sup> *wy ∂y*<sup>2</sup> � � �ð Þ <sup>A</sup> � <sup>B</sup> <sup>1</sup> *ρ*0 *∂δρ ∂x ∂ ρVy* � � *∂t* þ *∂ VxρVy* � � *∂x* þ *∂ VyρVy* � � *<sup>∂</sup><sup>y</sup>* ¼ � *<sup>∂</sup><sup>p</sup> ∂y* <sup>þ</sup> *<sup>η</sup>*∇<sup>2</sup> *Vy* þ 1 3 *η ∂ ∂y* ∇ � **V** <sup>þ</sup>*M*∇<sup>2</sup> *uy* <sup>þ</sup> ð Þ *<sup>L</sup>* <sup>þ</sup> *<sup>M</sup>* � *<sup>B</sup> <sup>∂</sup> ∂y* ∇ � **u** þ*R*<sup>1</sup> *∂*2 *wy <sup>∂</sup>x*<sup>2</sup> � <sup>2</sup> *∂*2 *wx <sup>∂</sup>x∂<sup>y</sup>* � *<sup>∂</sup>*<sup>2</sup> *wy ∂y*<sup>2</sup> � � þ *R*<sup>2</sup> *∂*2 *wx <sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> <sup>2</sup> *∂*2 *wy ∂x∂y* � *∂*2 *wx ∂y*<sup>2</sup> � � �ð Þ <sup>A</sup> � <sup>B</sup> <sup>1</sup> *ρ*0 *∂δρ ∂y ∂ux ∂t* þ *Vx ∂ux <sup>∂</sup><sup>x</sup>* <sup>þ</sup> *Vy ∂ux <sup>∂</sup><sup>y</sup>* <sup>¼</sup> *Vx* <sup>þ</sup> <sup>Γ</sup>**u**½*M*∇<sup>2</sup> *ux* <sup>þ</sup> ð Þ *<sup>L</sup>* <sup>þ</sup> *<sup>M</sup> <sup>∂</sup> <sup>∂</sup><sup>x</sup>* <sup>∇</sup> � **<sup>u</sup>**<sup>þ</sup> *R*1 *∂*2 *wx <sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> <sup>2</sup> *∂*2 *wy <sup>∂</sup>x∂<sup>y</sup>* � *<sup>∂</sup>wx ∂y*<sup>2</sup> � � � *R*<sup>2</sup> *∂*2 *wy <sup>∂</sup>x*<sup>2</sup> � <sup>2</sup> *∂*2 *wx <sup>∂</sup>x∂<sup>y</sup>* � *<sup>∂</sup>*<sup>2</sup> *wy ∂y*<sup>2</sup> � �# *∂uy ∂t* þ *Vx ∂uy <sup>∂</sup><sup>x</sup>* <sup>þ</sup> *Vy ∂uy <sup>∂</sup><sup>y</sup>* <sup>¼</sup> *Vy* <sup>þ</sup> <sup>Γ</sup>**u**½*M*∇<sup>2</sup> *uy* <sup>þ</sup> ð Þ *<sup>L</sup>* <sup>þ</sup> *<sup>M</sup> <sup>∂</sup> ∂y* ∇ � **u**þ *R*1 *∂*2 *wy <sup>∂</sup>x*<sup>2</sup> � <sup>2</sup> *∂*2 *wx <sup>∂</sup>x∂<sup>y</sup>* � *<sup>∂</sup>*<sup>2</sup> *wy ∂y*<sup>2</sup> � � þ *R*<sup>2</sup> *∂*2 *wx <sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> <sup>2</sup> *∂*2 *wy <sup>∂</sup>x∂<sup>y</sup>* � *<sup>∂</sup>*<sup>2</sup> *wx ∂y*<sup>2</sup> � �# *∂wx ∂t* þ *Vx ∂wx <sup>∂</sup><sup>x</sup>* <sup>þ</sup> *Vy ∂wx <sup>∂</sup><sup>y</sup>* <sup>¼</sup> <sup>Γ</sup>**w**½*K*1∇<sup>2</sup> *wx*þ *R*1 *∂*2 *ux <sup>∂</sup>x*<sup>2</sup> � <sup>2</sup> *∂*2 *uy ∂x∂y* � *∂*2 *ux ∂y*<sup>2</sup> � � þ *R*<sup>2</sup> *∂*2 *uy <sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> <sup>2</sup> *<sup>∂</sup>*<sup>2</sup> *ux ∂x∂y* � *∂*2 *uy ∂y*<sup>2</sup> � �# *∂wy ∂t* þ *Vx ∂wy <sup>∂</sup><sup>x</sup>* <sup>þ</sup> *Vy ∂wy <sup>∂</sup><sup>y</sup>* <sup>¼</sup> <sup>Γ</sup>**w**½*K*1∇<sup>2</sup> *wy*þ *R*1 *∂*2 *uy <sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> <sup>2</sup> *<sup>∂</sup>*<sup>2</sup> *ux ∂x∂y* � *∂*2 *uy ∂y*<sup>2</sup> � � � *R*<sup>2</sup> *∂*2 *ux <sup>∂</sup>x*<sup>2</sup> � <sup>2</sup> *∂*2 *uy ∂x∂y* � *∂*2 *ux ∂y*<sup>2</sup> � �# *p* ¼ 3 *kBT l* 3 *ρ ρ*0 þ *ρ*2 *ρ*0 <sup>2</sup> <sup>þ</sup> *<sup>ρ</sup>*<sup>3</sup> *ρ*0 3 � � 9 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>= >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>; (2)

in which *ui* denotes the phonon field, *wi* the phason field, *Vi* the fluid velocity field, *Cijkl* is the phonon elastic constant tensor, *Kijkl* phason elastic constant tensor, and *Rijkl* and *Rklij* are the phonon-phason coupling elastic constant tensors.

*A* and *B* are the constants describing density variation, *g* ¼ *ρ***V**, and *η* is the fluid viscosity, *kB* is the Boltzmann constant, *T* is the absolute temperature, and *l* � 10 � 100 nm is the characteristic size of the soft matter, respectively.

The initial and boundary value problem (3) of Eq. (2) can be solved by the finite

*<sup>h</sup>*<sup>2</sup> ,

*<sup>h</sup>*<sup>2</sup> ,

*∂xj*

*ξkl*,

� � (5)

9 >=

>;

, (4)

(6)

*<sup>∂</sup>x*<sup>2</sup> <sup>¼</sup> *ux*ð Þ� *<sup>i</sup>* <sup>þ</sup> 1, *<sup>j</sup>* <sup>2</sup>*ux*ð Þþ *<sup>i</sup>*, *<sup>j</sup> ux*ð Þ *<sup>i</sup>* � 1, *<sup>j</sup>*

*<sup>∂</sup>y*<sup>2</sup> <sup>¼</sup> *ux*ð Þ� *<sup>i</sup>*, *<sup>j</sup>* <sup>þ</sup> <sup>1</sup> <sup>2</sup>*ux*ð Þþ *<sup>i</sup>*, *<sup>j</sup> ux*ð Þ *<sup>i</sup>*, *<sup>j</sup>* � <sup>1</sup>

*<sup>τ</sup> :*

so determine the phonon and phason displacement fields and fluid phonon

*∂Vi ∂xj* þ *∂Vj ∂xi*

*ij* ¼ �*pδij* <sup>þ</sup> <sup>2</sup>*<sup>η</sup>* \_

*∂ui ∂xj* þ *∂uj ∂xi* � �, *wij* <sup>¼</sup> *<sup>∂</sup>wi*

difference method to solve the boundary value problem (**Figure 2**), e.g.,

*<sup>∂</sup><sup>t</sup>* <sup>¼</sup> *ux*ð Þ� *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> *ux*ð Þ*<sup>k</sup>*

velocity filed, then the phonon and phason strain tensors.

*<sup>ε</sup>ij* <sup>¼</sup> <sup>1</sup> 2

> \_ *<sup>ξ</sup>ij* <sup>¼</sup> <sup>1</sup> 2

*pij* ¼ �*pδij* þ *σ*<sup>0</sup>

*σij* ¼ *Cijklεik* þ R*ijklwkl*, *Hij* ¼ *Kijklwij* þ R*klijεkl*,

and the fluid phonon deformation rate tensor

and according to constitutive laws

*The scheme of grid of the difference for a part of the specimen.*

**Figure 2.**

**5**

*∂*2 *ux*

*Probe on Rupture Theory of Soft-Matter Quasicrystals DOI: http://dx.doi.org/10.5772/intechopen.89813*

> *∂*2 *ux*

*∂ux*

## **4. An example of stress analysis of soft-matter quasicrystals**

With Eq. (2), we can carry out a stress analysis of some fundamental specimens with crack of soft-matter quasicrystals; we here give only a computational example as shown in **Figure 1**.

If we want to obtain further information on deformation and motion of the material, we must solve the equations under appropriate initial and boundary conditions. To solve the problem, a specimen made by the matter should be optioned, which is subjected to some initial and boundary conditions. Here, the corresponding conditions of the specimen shown in **Figure 1** are as follows:

$$\begin{aligned} \mathbf{t} &= \mathbf{0}: V\_x = V\_y = \mathbf{0}, \mathbf{u}\_x = \mathbf{u}\_y = \mathbf{0}, \mathbf{w}\_x = w\_y = \mathbf{0}, \ p = p\_0; \\ \mathbf{y} &= \pm H, |\mathbf{x}| < \mathbf{W}: V\_x = V\_y = \mathbf{0}, \sigma\_{\mathbf{y}\mathbf{y}} = \sigma\_0 \mathbf{f}(\mathbf{t}), \sigma\_{\mathbf{y}\mathbf{x}} = \mathbf{0}, H\_{\mathbf{y}\mathbf{y}} = H\_{\mathbf{y}\mathbf{x}} = \mathbf{0}, p = p\_0; \\ \mathbf{x} &= \pm W, |\mathbf{y}| < \mathbf{H}: V\_x = V\_{\mathbf{y}} = \mathbf{0}, \sigma\_{\mathbf{x}\mathbf{x}} = \sigma\_{\mathbf{xy}} = \mathbf{0}, H\_{\mathbf{x}\mathbf{x}} = H\_{\mathbf{xy}} = \mathbf{0}, p = p\_0; \\ \mathbf{y} &= \mathbf{0}, |\mathbf{x}| < a : V\_x = V\_{\mathbf{y}} = \mathbf{0}, \sigma\_{\mathbf{y}\mathbf{y}} = \sigma\_{\mathbf{x}\mathbf{x}} = \mathbf{0}, H\_{\mathbf{y}\mathbf{y}} = H\_{\mathbf{yx}} = \mathbf{0}, p = p\_0 \end{aligned} \tag{3}$$

In the present computation we take, 2*H* ¼ 0*:*01m, 2*W* ¼ 0*:*01m, 2*a* ¼ 0*:*0024m, *<sup>σ</sup>*<sup>0</sup> <sup>¼</sup> <sup>0</sup>*:*01MPa, *<sup>ρ</sup>*<sup>0</sup> <sup>¼</sup> <sup>1</sup>*:*<sup>5</sup> � 103 kg*=*m3,*<sup>η</sup>* <sup>¼</sup> <sup>0</sup>*:*1 Pa � s or 0 ð Þ *:*2 Pa � <sup>s</sup> ,*<sup>ζ</sup>* <sup>¼</sup> 0,*<sup>L</sup>* <sup>¼</sup> 10MPa, *<sup>M</sup>* <sup>¼</sup> 4MPa,*K*<sup>1</sup> <sup>¼</sup> <sup>0</sup>*:*5 L, *<sup>R</sup>* <sup>¼</sup> *<sup>R</sup>*<sup>1</sup> <sup>¼</sup> <sup>0</sup>*:*04M or 0 ð Þ *:*06M , *<sup>R</sup>*<sup>2</sup> <sup>¼</sup> 0, <sup>Γ</sup>*<sup>u</sup>* <sup>¼</sup> <sup>4</sup>*:*<sup>8</sup> � <sup>10</sup>�<sup>17</sup> <sup>m</sup><sup>3</sup> � <sup>s</sup>*=*kg, <sup>Γ</sup>*<sup>w</sup>* <sup>¼</sup> <sup>4</sup>*:*<sup>8</sup> � <sup>10</sup>�<sup>19</sup> <sup>m</sup><sup>3</sup> � <sup>s</sup>*=*kg, A � <sup>0</sup>*:*2MPa, B � <sup>0</sup>*:*2MPa, and *<sup>p</sup>*<sup>0</sup> denotes 1 atm.

**Figure 1.** *Specimen of soft-matter quasicrystals of 5- and 10-fold symmetries with a Griffith crack under tension.*

*Probe on Rupture Theory of Soft-Matter Quasicrystals DOI: http://dx.doi.org/10.5772/intechopen.89813*

*A* and *B* are the constants describing density variation, *g* ¼ *ρ***V**, and *η* is the fluid viscosity, *kB* is the Boltzmann constant, *T* is the absolute temperature, and

With Eq. (2), we can carry out a stress analysis of some fundamental specimens with crack of soft-matter quasicrystals; we here give only a computational example

If we want to obtain further information on deformation and motion of the material, we must solve the equations under appropriate initial and boundary conditions. To solve the problem, a specimen made by the matter should be optioned, which is subjected to some initial and boundary conditions. Here, the corresponding

t ¼ 0 : *Vx* ¼ *Vy* ¼ 0, u*<sup>x</sup>* ¼ u*<sup>y</sup>* ¼ 0, w*<sup>x</sup>* ¼ *wy* ¼ 0, *p* ¼ *p*0; *y* ¼ �*H*, j j *x* <*W* : *Vx* ¼ *Vy* ¼ 0, *σyy* ¼ *σ*0*f t*ð Þ, *σyx* ¼ 0, *Hyy* ¼ *Hyx* ¼ 0, *p* ¼ *p*0;

In the present computation we take, 2*H* ¼ 0*:*01m, 2*W* ¼ 0*:*01m, 2*a* ¼ 0*:*0024m, *<sup>σ</sup>*<sup>0</sup> <sup>¼</sup> <sup>0</sup>*:*01MPa, *<sup>ρ</sup>*<sup>0</sup> <sup>¼</sup> <sup>1</sup>*:*<sup>5</sup> � 103 kg*=*m3,*<sup>η</sup>* <sup>¼</sup> <sup>0</sup>*:*1 Pa � s or 0 ð Þ *:*2 Pa � <sup>s</sup> ,*<sup>ζ</sup>* <sup>¼</sup> 0,*<sup>L</sup>* <sup>¼</sup> 10MPa, *<sup>M</sup>* <sup>¼</sup> 4MPa,*K*<sup>1</sup> <sup>¼</sup> <sup>0</sup>*:*5 L, *<sup>R</sup>* <sup>¼</sup> *<sup>R</sup>*<sup>1</sup> <sup>¼</sup> <sup>0</sup>*:*04M or 0 ð Þ *:*06M , *<sup>R</sup>*<sup>2</sup> <sup>¼</sup> 0, <sup>Γ</sup>*<sup>u</sup>* <sup>¼</sup> <sup>4</sup>*:*<sup>8</sup> � <sup>10</sup>�<sup>17</sup> <sup>m</sup><sup>3</sup> � <sup>s</sup>*=*kg, <sup>Γ</sup>*<sup>w</sup>* <sup>¼</sup> <sup>4</sup>*:*<sup>8</sup> � <sup>10</sup>�<sup>19</sup> <sup>m</sup><sup>3</sup> � <sup>s</sup>*=*kg, A � <sup>0</sup>*:*2MPa, B � <sup>0</sup>*:*2MPa, and *<sup>p</sup>*<sup>0</sup> denotes 1 atm.

(3)

*x* ¼ �*W*, j j *y* < *H* : *Vx* ¼ *Vy* ¼ 0, *σxx* ¼ *σxy* ¼ 0, *Hxx* ¼ *Hxy* ¼ 0, *p* ¼ *p*0; *y* ¼ 0, j j *x* <*a* : *Vx* ¼ *Vy* ¼ 0, *σyy* ¼ *σyx* ¼ 0, *Hyy* ¼ *Hyx* ¼ 0, *p* ¼ *p*<sup>0</sup>

*Specimen of soft-matter quasicrystals of 5- and 10-fold symmetries with a Griffith crack under tension.*

*l* � 10 � 100 nm is the characteristic size of the soft matter, respectively.

**4. An example of stress analysis of soft-matter quasicrystals**

conditions of the specimen shown in **Figure 1** are as follows:

as shown in **Figure 1**.

*Fracture Mechanics Applications*

**Figure 1.**

**4**

The initial and boundary value problem (3) of Eq. (2) can be solved by the finite difference method to solve the boundary value problem (**Figure 2**), e.g.,

$$\begin{aligned} \frac{\partial^2 u\_\mathbf{x}}{\partial \mathbf{x}^2} &= \frac{u\_\mathbf{x}(i+1,j) - 2u\_\mathbf{x}(i,j) + u\_\mathbf{x}(i-1,j)}{h^2}, \\\frac{\partial^2 u\_\mathbf{x}}{\partial y^2} &= \frac{u\_\mathbf{x}(i,j+1) - 2u\_\mathbf{x}(i,j) + u\_\mathbf{x}(i,j-1)}{h^2}, \\\frac{\partial u\_\mathbf{x}}{\partial t} &= \frac{u\_\mathbf{x}(k+1) - u\_\mathbf{x}(k)}{\tau}. \end{aligned}$$

so determine the phonon and phason displacement fields and fluid phonon velocity filed, then the phonon and phason strain tensors.

$$\varepsilon\_{i\bar{j}} = \frac{1}{2} \left( \frac{\partial u\_i}{\partial \mathbf{x}\_{\bar{j}}} + \frac{\partial u\_{\bar{j}}}{\partial \mathbf{x}\_{i}} \right), w\_{i\bar{j}} = \frac{\partial w\_{i}}{\partial \mathbf{x}\_{\bar{j}}}, \tag{4}$$

and the fluid phonon deformation rate tensor

$$\dot{\xi}\_{ij} = \frac{1}{2} \left( \frac{\partial V\_i}{\partial \mathbf{x}\_j} + \frac{\partial V\_j}{\partial \mathbf{x}\_i} \right) \tag{5}$$

and according to constitutive laws

$$\begin{aligned} \sigma\_{\vec{\imath}\dot{\jmath}} &= \mathbf{C}\_{\ddot{\jmath}kl}\varepsilon\_{ik} + \mathbf{R}\_{\ddot{\jmath}kl}w\_{kl}, \\ H\_{\vec{\imath}\dot{\jmath}} &= K\_{\ddot{\jmath}kl}w\_{\dddot{\jmath}} + \mathbf{R}\_{kl\dot{\jmath}}\varepsilon\_{kl}, \\ p\_{\dddot{\jmath}} &= -p\delta\_{\ddot{\jmath}} + \sigma\_{\dddot{\jmath}}' = -p\delta\_{\ddot{\jmath}} + 2\eta\dot{\xi}\_{kl}, \end{aligned} \tag{6}$$

**Figure 2.** *The scheme of grid of the difference for a part of the specimen.*

we obtain the phonon stresses *σij* and fluid phonon stresses *pij*, respectively, in which recall *Cijkl* the phonon elastic constant tensor, *Kijkl* phason elastic constant tensor, *Rijkl* and *Rklij* the phonon-phason coupling elastic constant tensors, etc., refer to [24].

Due to the complexity of the equations, the computation is complex too. For the dynamic problems, i.e., there are manmade damping terms *θρ*, *θρVx*, and so on, the iterative computation is easily stable, and for the static problems, we still take the iterative computation; however, the choosing of the manmade damping coefficient *θ* depends upon experience.

The computational results are mostly dependent on the ratio value of *σ*0*=p*0, i.e., the ratio of amplitudes of the phonon stress and fluid phonon stress apart from other factors.

## **5. Significance of fluid stress to crack initiation of growth and crack propagation**

In the crack and fracture of brittle or quasibrittle failure of structural materials, the stress analysis is a basic task, from which one can determine the stress intensity factor, and then use the Griffith-Irwin criterion to analyze the crack stability/instability. For the Mode I crack, the tensile stress *σyy* x, y, t is the most important, which leads to the crack surface opening and so the crack propagation.

intensity factor). *K*<sup>1</sup>

**Figure 4.**

because *K*<sup>1</sup>

*K*1

**7**

phonon denotes the phonon stress intensity factor and *K*<sup>1</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *π*ð Þ *x* � *a*<sup>0</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *π*ð Þ *x* � *a*<sup>0</sup>

The computational results are dependent upon the ratio *σ*0*=p*0, while the fluid stress

fluid < 0 leads to crack surface closing. The total stress intensity factor is denoted by

total is smaller than *K*<sup>1</sup>

*K*<sup>1</sup> is the most often used in engineering design parameter in fracture mechanics and, hence, must be understood if we are to design fracture tolerant materials used in bridges, buildings, aircraft, or even bells. Polishing cannot detect a crack. Typically, if a crack can be seen, it is very close to the critical stress state predicted by the stress intensity factor. The magnitude of*K*~<sup>1</sup> depends on sample geometry, the size and location of the crack, and

phonon>0 leads to crack surface opening, and according to our computation,

phonon <sup>þ</sup> *<sup>K</sup>*<sup>1</sup>

<sup>p</sup> *<sup>σ</sup>yy*ð Þ x, 0, *<sup>t</sup>* (7)

<sup>p</sup> *pyy*ð Þ x, 0, *<sup>t</sup>* (8)

fluid*:* (9)

<sup>p</sup> *<sup>σ</sup>*<sup>0</sup> (10)

total <sup>¼</sup> *KIC*ð Þ *<sup>T</sup> :* (11)

fluid, which is also a function of time in general,

phonon.

denotes the fluid phonon stress intensity factor which can be defined as follows:

*x*!*a*<sup>þ</sup> 0

*K*1

*Probe on Rupture Theory of Soft-Matter Quasicrystals DOI: http://dx.doi.org/10.5772/intechopen.89813*

*K*1

intensity factor is expressed by *K*<sup>1</sup>

Therefore, the value of *K*<sup>1</sup>

**6. Possible criterion of rupture**

For soft matter, there is fracture criterion:

phonon <sup>¼</sup> lim

*Normal stress pyy*ð Þ *x*, *0*, *t of fluid field at the point A of specimen versus time.*

fluid <sup>¼</sup> lim *x*!*a*<sup>þ</sup> 0

*K*1

the magnitude and the modal distribution of loads on the material.

The normalized elastic dynamic stress intensity factor is defined as

*K*1

*<sup>K</sup>*~1ðÞ¼ *<sup>t</sup> <sup>K</sup>*1ð Þ*<sup>t</sup> <sup>=</sup>* ffiffiffiffiffiffiffiffi

*πa*<sup>0</sup>

total <sup>¼</sup> *<sup>K</sup>*<sup>1</sup>

*fluid*

After our computation, in soft matter including soft-matter quasicrystals, apart from the phonon stress *<sup>σ</sup>yy* x, y, t , there is the fluid phonon stress *pyy* x, y, t which is pressure and leads to crack closing (**Figures 3** and **4**).

In principle, the Griffith theory holds for describing brittle and quasi-brittle rupture of structural materials (or engineering materials). However, for soft matter including soft-matter quasicrystals, there is fundamental difference with the structural materials. The key reason about this is the existence of fluid effect; in terminology of soft-matter quasicrystal study, it is also called the existence of fluid phonon. According to our analysis, the effect of fluid stress intensity factor is opposite to the elastic stress intensity factor (or by using the terminology of soft-matter quasicrystals study, the elastic stress intensity factor is also called the phonon stress

**Figure 3.** *Normal stress σyy*ð Þ *x*, *0*, *t of phonon field at the point A of specimen versus time.*

*Probe on Rupture Theory of Soft-Matter Quasicrystals DOI: http://dx.doi.org/10.5772/intechopen.89813*

we obtain the phonon stresses *σij* and fluid phonon stresses *pij*, respectively, in which recall *Cijkl* the phonon elastic constant tensor, *Kijkl* phason elastic constant tensor, *Rijkl* and *Rklij* the phonon-phason coupling elastic constant tensors, etc., refer to [24].

Due to the complexity of the equations, the computation is complex too. For the dynamic problems, i.e., there are manmade damping terms *θρ*, *θρVx*, and so on, the iterative computation is easily stable, and for the static problems, we still take the iterative computation; however, the choosing of the manmade damping coefficient

The computational results are mostly dependent on the ratio value of *σ*0*=p*0, i.e.,

In the crack and fracture of brittle or quasibrittle failure of structural materials, the stress analysis is a basic task, from which one can determine the stress intensity factor, and then use the Griffith-Irwin criterion to analyze the crack stability/insta-

After our computation, in soft matter including soft-matter quasicrystals, apart

In principle, the Griffith theory holds for describing brittle and quasi-brittle rupture of structural materials (or engineering materials). However, for soft matter including soft-matter quasicrystals, there is fundamental difference with the structural materials. The key reason about this is the existence of fluid effect; in terminology of soft-matter quasicrystal study, it is also called the existence of fluid phonon. According to our analysis, the effect of fluid stress intensity factor is opposite to the elastic stress intensity factor (or by using the terminology of soft-matter quasicrystals study, the elastic stress intensity factor is also called the phonon stress

, there is the fluid phonon stress *pyy* x, y, t

is the most important,

which

the ratio of amplitudes of the phonon stress and fluid phonon stress apart from

**5. Significance of fluid stress to crack initiation of growth and crack**

which leads to the crack surface opening and so the crack propagation.

bility. For the Mode I crack, the tensile stress *σyy* x, y, t

is pressure and leads to crack closing (**Figures 3** and **4**).

*Normal stress σyy*ð Þ *x*, *0*, *t of phonon field at the point A of specimen versus time.*

*θ* depends upon experience.

*Fracture Mechanics Applications*

from the phonon stress *σyy* x, y, t

other factors.

**Figure 3.**

**6**

**propagation**

**Figure 4.** *Normal stress pyy*ð Þ *x*, *0*, *t of fluid field at the point A of specimen versus time.*

intensity factor). *K*<sup>1</sup> phonon denotes the phonon stress intensity factor and *K*<sup>1</sup> *fluid* denotes the fluid phonon stress intensity factor which can be defined as follows:

$$K\_1^{\text{phonon}} = \lim\_{\mathbf{x} \to a\_0^+} \sqrt{\pi(\mathbf{x} - a\_0)} \sigma\_{\mathcal{Y}^\mathbf{y}}(\mathbf{x}, \mathbf{0}, t) \tag{7}$$

$$K\_1^{\text{fluid}} = \lim\_{\mathbf{x} \to a\_0^+} \sqrt{\pi(\mathbf{x} - a\_0)} p\_{\mathcal{\mathcal{Y}}}(\mathbf{x}, \mathbf{0}, t) \tag{8}$$

The computational results are dependent upon the ratio *σ*0*=p*0, while the fluid stress intensity factor is expressed by *K*<sup>1</sup> fluid, which is also a function of time in general, because *K*<sup>1</sup> phonon>0 leads to crack surface opening, and according to our computation, *K*1 fluid < 0 leads to crack surface closing. The total stress intensity factor is denoted by

$$K\_1^{\text{total}} = K\_1^{\text{phonon}} + K\_1^{\text{fluid}}.\tag{9}$$

Therefore, the value of *K*<sup>1</sup> total is smaller than *K*<sup>1</sup> phonon.

*K*<sup>1</sup> is the most often used in engineering design parameter in fracture mechanics and, hence, must be understood if we are to design fracture tolerant materials used in bridges, buildings, aircraft, or even bells. Polishing cannot detect a crack. Typically, if a crack can be seen, it is very close to the critical stress state predicted by the stress intensity factor. The magnitude of*K*~<sup>1</sup> depends on sample geometry, the size and location of the crack, and the magnitude and the modal distribution of loads on the material.

The normalized elastic dynamic stress intensity factor is defined as

$$
\tilde{K}\_1(t) = K\_1(t) / \sqrt{\pi a\_0} \sigma\_0 \tag{10}
$$

## **6. Possible criterion of rupture**

For soft matter, there is fracture criterion:

$$K\_1^{\text{total}} = K\_{IC}(T). \tag{11}$$

The Griffith theory can be extended to soft matter. If *K*<sup>1</sup>

**7. The references for other soft matter**

*Probe on Rupture Theory of Soft-Matter Quasicrystals DOI: http://dx.doi.org/10.5772/intechopen.89813*

out of the soft-matter quasicrystals may be set up.

**8. Conclusion and discussion**

**Acknowledgements**

through grant 11272053.

**Conflict of interest**

**9**

The authors declare no conflict of interest.

*K*1

surface to be opening, and crack propagate leads to fracture; otherwise

totalð Þ*<sup>t</sup>* <sup>&</sup>lt;*KIC*ð Þ *<sup>T</sup>* then crack surface to be closing, and crack cannot propagate, the material is in safe. The conclusion is that the physical state around crack tip in soft matter quasicrystals is dependent on the competition of phonon stress and fluid stress.

The example given in previous section is only the most preliminary discussion on the probe of rupture of soft-matter quasicrystals. It does not mean any in-depth and exact theory of the problem. The significance of the discussion lies in providing a possible drawing for other categories of soft matter in the quantitative study of rupture problems. The formulation of the present study may be referenced by other branches of soft matter, in which the constitutive laws for themselves are needed, and phason field can be excluded for non-quasicrystalline materials, and so on. With these considerations, the rupture theory for the other categories of soft matter

From **Figure 5**, we also can see that the real value of fluid DSIF is negative; thus, we can say it causes the crack closed. When comparing (a) with (b), the absolute value of fluid DSIF is larger than the elastic ones; thus, the crack cannot propagate. Then, we can conclude that in the quasicrystal of soft matter, the fluid effect is important, and we should do the study under the hydrodynamics theory.

The equation set of generalized dynamics of soft-matter quasicrystals is complete; it is possible to provide the hydrodynamics study on the mass distribution, deformation, and motion, including rupture analysis of the new phase. In particular, the dynamics stress intensity factor (DSIF) was evaluated, and the computation presents highly stability. In the results, there are two regimes: one is elastic DSIF and the other is fluid DSIF. The former leads to crack surface opening and the latter leads to crack surface closing. We can conclude that whether or not the crack open or close depends on the competition between the elastic DSIF and fluid DSIF. Of course, the theoretical analysis is anxious for the experimental verification currently, which is our attempt work in future. The above introducing is rough and preliminary, in fact, around the crack tip, due to the stress concentration and high stress grad, nonelastic stress appears, the Griffith theory will not be effective, i.e., the theory based on the stress intensity factor will not be valid, one must carry out nonlinear analysis, e.g. the theory based on the crack tip opening displacement, in which the size of the plastic zone near the crack tip is also considered, refer to Fan [24], which may help us to overcome the difficulty mentioned above. However this is a very difficult question which has not been solved so far.

This work is supported by the National Natural Science Foundation of China

totalð Þ*<sup>t</sup>* <sup>&</sup>gt;*KIC*ð Þ *<sup>T</sup>* then crack

*For soft-matter quasicrystals with 5- and 10-fold symmetries: (a) normalized phonon (elastic) DSIF versus time; (b) normalized fluid DSIF versus time; and (c) normalized total DSIF versus time.*

*Probe on Rupture Theory of Soft-Matter Quasicrystals DOI: http://dx.doi.org/10.5772/intechopen.89813*

The Griffith theory can be extended to soft matter. If *K*<sup>1</sup> totalð Þ*<sup>t</sup>* <sup>&</sup>gt;*KIC*ð Þ *<sup>T</sup>* then crack surface to be opening, and crack propagate leads to fracture; otherwise *K*1 totalð Þ*<sup>t</sup>* <sup>&</sup>lt;*KIC*ð Þ *<sup>T</sup>* then crack surface to be closing, and crack cannot propagate, the material is in safe. The conclusion is that the physical state around crack tip in soft matter quasicrystals is dependent on the competition of phonon stress and fluid stress.

## **7. The references for other soft matter**

The example given in previous section is only the most preliminary discussion on the probe of rupture of soft-matter quasicrystals. It does not mean any in-depth and exact theory of the problem. The significance of the discussion lies in providing a possible drawing for other categories of soft matter in the quantitative study of rupture problems. The formulation of the present study may be referenced by other branches of soft matter, in which the constitutive laws for themselves are needed, and phason field can be excluded for non-quasicrystalline materials, and so on. With these considerations, the rupture theory for the other categories of soft matter out of the soft-matter quasicrystals may be set up.

## **8. Conclusion and discussion**

From **Figure 5**, we also can see that the real value of fluid DSIF is negative; thus, we can say it causes the crack closed. When comparing (a) with (b), the absolute value of fluid DSIF is larger than the elastic ones; thus, the crack cannot propagate. Then, we can conclude that in the quasicrystal of soft matter, the fluid effect is important, and we should do the study under the hydrodynamics theory.

The equation set of generalized dynamics of soft-matter quasicrystals is complete; it is possible to provide the hydrodynamics study on the mass distribution, deformation, and motion, including rupture analysis of the new phase. In particular, the dynamics stress intensity factor (DSIF) was evaluated, and the computation presents highly stability. In the results, there are two regimes: one is elastic DSIF and the other is fluid DSIF. The former leads to crack surface opening and the latter leads to crack surface closing. We can conclude that whether or not the crack open or close depends on the competition between the elastic DSIF and fluid DSIF. Of course, the theoretical analysis is anxious for the experimental verification currently, which is our attempt work in future.

The above introducing is rough and preliminary, in fact, around the crack tip, due to the stress concentration and high stress grad, nonelastic stress appears, the Griffith theory will not be effective, i.e., the theory based on the stress intensity factor will not be valid, one must carry out nonlinear analysis, e.g. the theory based on the crack tip opening displacement, in which the size of the plastic zone near the crack tip is also considered, refer to Fan [24], which may help us to overcome the difficulty mentioned above. However this is a very difficult question which has not been solved so far.

## **Acknowledgements**

This work is supported by the National Natural Science Foundation of China through grant 11272053.

## **Conflict of interest**

The authors declare no conflict of interest.

**Figure 5.**

*Fracture Mechanics Applications*

**8**

*For soft-matter quasicrystals with 5- and 10-fold symmetries: (a) normalized phonon (elastic) DSIF versus*

*time; (b) normalized fluid DSIF versus time; and (c) normalized total DSIF versus time.*

## **A. Appendix**

## **A.1 The final governing equations of generalized dynamics of soft-matter quasicrystals with 10-fold symmetry in three dimension**

In the text, Eq. (2) is the planar field form only. In book [24], it provides an explicit form of the equations and lists below soft-matter quasicrystals with 10-fold symmetry:

*∂ρ ∂t* þ ∇ � ð Þ¼ *ρ***V** 0 *<sup>∂</sup>*ð Þ *<sup>ρ</sup>Vx ∂t* þ *<sup>∂</sup>*ð Þ *VxρVx ∂x* þ *∂ VyρVx* � � *∂y* þ *<sup>∂</sup>*ð Þ *VzρVx <sup>∂</sup><sup>z</sup>* ¼ � *<sup>∂</sup><sup>p</sup> <sup>∂</sup><sup>x</sup>* <sup>þ</sup> *<sup>η</sup>*∇<sup>2</sup> *Vx* þ 1 3 *η ∂ <sup>∂</sup><sup>x</sup>* <sup>∇</sup> � **<sup>V</sup>** þ *C*<sup>66</sup> *∂*2 *<sup>∂</sup>y*<sup>2</sup> <sup>þ</sup> *<sup>C</sup>*<sup>44</sup> *∂*2 *∂z*<sup>2</sup> � �*ux* <sup>þ</sup> ð Þ *<sup>C</sup>*<sup>12</sup> <sup>þ</sup> *<sup>C</sup>*<sup>66</sup> *∂*2 *uy ∂x∂y* þ ð Þ *C*<sup>13</sup> þ *C*<sup>44</sup> � *C*<sup>11</sup> *∂*2 *uz ∂x∂z* <sup>þ</sup> ð Þ *<sup>C</sup>*<sup>11</sup> � *<sup>B</sup> <sup>∂</sup> <sup>∂</sup><sup>x</sup>* <sup>∇</sup> � **<sup>u</sup>** <sup>þ</sup>*<sup>R</sup> <sup>∂</sup> <sup>∂</sup><sup>x</sup>* <sup>∇</sup><sup>1</sup> � **<sup>w</sup>** � *<sup>R</sup> <sup>∂</sup> ∂y ∂wx <sup>∂</sup><sup>y</sup>* � *<sup>∂</sup>wy ∂x* � � � ð Þ *<sup>A</sup>* � *<sup>B</sup>* <sup>1</sup> *ρ*0 *∂δρ ∂x ∂ ρVy* � � *∂t* þ *∂ VxρVy* � � *∂x* þ *∂ VyρVy* � � *∂y* þ *∂ VzρVy* � � *<sup>∂</sup><sup>z</sup>* ¼ � *<sup>∂</sup><sup>p</sup> ∂y* <sup>þ</sup> *<sup>η</sup>*∇<sup>2</sup> *Vy* þ 1 3 *η ∂ ∂y* ∇ � **V** þð Þ *C*<sup>12</sup> þ *C*<sup>66</sup> *∂*2 *ux ∂x∂y* þ *C*<sup>66</sup> *∂*2 *<sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> *<sup>C</sup>*<sup>11</sup> *∂*2 *<sup>∂</sup>y*<sup>2</sup> <sup>þ</sup> *<sup>C</sup>*<sup>44</sup> *∂*2 *∂z*<sup>2</sup> � �*uy* <sup>þ</sup> ð Þ *<sup>C</sup>*<sup>13</sup> <sup>þ</sup> *<sup>C</sup>*<sup>44</sup> *∂*2 *uz ∂y∂z* <sup>þ</sup> ð Þ *<sup>C</sup>*<sup>11</sup> � *<sup>B</sup> <sup>∂</sup> ∂y* ∇ � **u** �*<sup>R</sup> <sup>∂</sup> ∂x ∂wx <sup>∂</sup><sup>y</sup>* � *<sup>∂</sup>wy ∂x* � � � *<sup>R</sup> <sup>∂</sup> ∂y* <sup>∇</sup><sup>1</sup> � **<sup>w</sup>** � ð Þ *<sup>A</sup>* � *<sup>B</sup>* <sup>1</sup> *ρ*0 *∂δρ ∂y <sup>∂</sup>*ð Þ *<sup>ρ</sup>Vz ∂t* þ *<sup>∂</sup>*ð Þ *VxρVz ∂x* þ *∂ VyρVz* � � *∂y* þ *<sup>∂</sup>*ð Þ *VzρVz <sup>∂</sup><sup>z</sup>* ¼ � *<sup>∂</sup><sup>p</sup> <sup>∂</sup><sup>z</sup>* <sup>þ</sup> *<sup>η</sup>*∇<sup>2</sup> *Vz* þ 1 3 *η ∂ ∂z* ∇ � **V** þ *C*<sup>44</sup> *∂*2 *<sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> *<sup>C</sup>*<sup>44</sup> *∂*2 *<sup>∂</sup>y*<sup>2</sup> <sup>þ</sup> ð Þ *<sup>C</sup>*<sup>33</sup> � *<sup>C</sup>*<sup>13</sup> � *<sup>C</sup>*<sup>44</sup> *∂*2 *∂z*<sup>2</sup> � �*uz* <sup>þ</sup> ð Þ *<sup>C</sup>*<sup>13</sup> <sup>þ</sup> *<sup>C</sup>*<sup>44</sup> � <sup>B</sup> *<sup>∂</sup> ∂z* <sup>∇</sup> � **<sup>u</sup>** � ð Þ *<sup>A</sup>* � *<sup>B</sup>* <sup>1</sup> *ρ*0 *∂δρ ∂z ∂ux ∂t* þ *Vx ∂ux <sup>∂</sup><sup>x</sup>* <sup>þ</sup> *Vy ∂ux ∂y* þ *Vz ∂ux ∂z* ¼ *Vx* þ Γ**<sup>u</sup>** " *C*<sup>11</sup> *∂*2 *<sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> *<sup>C</sup>*<sup>66</sup> *∂*2 *<sup>∂</sup>y*<sup>2</sup> <sup>þ</sup> *<sup>C</sup>*<sup>44</sup> *∂*2 *∂z*<sup>2</sup> � �*ux* <sup>þ</sup> ð Þ *<sup>C</sup>*<sup>12</sup> <sup>þ</sup> *<sup>C</sup>*<sup>66</sup> *∂*2 *uy ∂x∂y* þð Þ *C*<sup>13</sup> þ *C*<sup>44</sup> *∂*2 *uz ∂x∂z* <sup>þ</sup> *<sup>R</sup> <sup>∂</sup> <sup>∂</sup><sup>x</sup>* <sup>∇</sup><sup>1</sup> � **<sup>w</sup>** � *<sup>R</sup> <sup>∂</sup> ∂y ∂wx <sup>∂</sup><sup>y</sup>* � *<sup>∂</sup>wy ∂x* � �# *∂uy ∂t* þ *Vx ∂uy <sup>∂</sup><sup>x</sup>* <sup>þ</sup> *Vy ∂uy ∂y* þ *Vz ∂uy ∂z* ¼ *Vy* þ Γ**<sup>u</sup>** " ð Þ *C*<sup>12</sup> þ *C*<sup>66</sup> *∂*2 *ux ∂x∂y* þ *C*<sup>66</sup> *∂*2 *<sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> *<sup>C</sup>*<sup>11</sup> *∂*2 *<sup>∂</sup>y*<sup>2</sup> <sup>þ</sup> *<sup>C</sup>*<sup>44</sup> *∂*2 *∂z*<sup>2</sup> � �*uy* þð Þ *C*<sup>13</sup> þ *C*<sup>44</sup> *∂*2 *uz ∂y∂z* � *<sup>R</sup> <sup>∂</sup> ∂x ∂wx <sup>∂</sup><sup>y</sup>* � *<sup>∂</sup>wy ∂x* � � � *<sup>R</sup> <sup>∂</sup> ∂y* ∇<sup>1</sup> � **w** # *∂uz ∂t* þ *Vx ∂uz <sup>∂</sup><sup>x</sup>* <sup>þ</sup> *Vy ∂uz ∂y* þ *Vz ∂uz <sup>∂</sup><sup>z</sup>* <sup>¼</sup> *Vz* <sup>þ</sup> <sup>Γ</sup>**<sup>u</sup>** " ð Þ *C*<sup>13</sup> þ *C*<sup>44</sup> *∂*2 *ux ∂x∂z* þ *∂*2 *uy ∂y∂z* � � þ *C*<sup>44</sup> *∂*2 *<sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> *<sup>C</sup>*<sup>44</sup> *∂*2 *<sup>∂</sup>y*<sup>2</sup> <sup>þ</sup> *<sup>C</sup>*<sup>33</sup> *∂*2 *∂z*<sup>2</sup> � �*uz* # *∂wx ∂t* þ *Vx ∂wx <sup>∂</sup><sup>x</sup>* <sup>þ</sup> *Vy ∂wx ∂y* þ *Vz ∂wx ∂z* <sup>¼</sup> <sup>Γ</sup>**<sup>w</sup>** *<sup>K</sup>*1∇<sup>2</sup> <sup>1</sup>*wx* þ *K*<sup>4</sup> *∂*2 *wx <sup>∂</sup>z*<sup>2</sup> <sup>þ</sup> *<sup>K</sup>*<sup>2</sup> *∂ ∂y ∂ <sup>∂</sup><sup>x</sup>* � *<sup>∂</sup> ∂y* � �*wy* <sup>þ</sup> *<sup>R</sup> <sup>∂</sup> ∂x ∂ux <sup>∂</sup><sup>x</sup>* � *<sup>∂</sup>uy ∂y* � � � *<sup>R</sup> <sup>∂</sup> ∂y ∂ux ∂y* þ *∂uy ∂x* � � �� *∂wy ∂t* þ *Vx ∂wy <sup>∂</sup><sup>x</sup>* <sup>þ</sup> *Vy ∂wy ∂y Vz ∂wy ∂z* <sup>¼</sup> <sup>Γ</sup>**<sup>w</sup>** *<sup>K</sup>*1∇<sup>2</sup> <sup>1</sup>*wy* þ *K*<sup>4</sup> *∂*2 *wy <sup>∂</sup>z*<sup>2</sup> <sup>þ</sup> *<sup>R</sup> <sup>∂</sup> ∂x ∂ux ∂y* þ *∂uy ∂x* � � <sup>þ</sup> *<sup>R</sup> <sup>∂</sup> ∂y ∂ux <sup>∂</sup><sup>x</sup>* � *<sup>∂</sup>uy ∂y* � � � � *p* ¼ *f*ð Þ¼ *ρ* 3 *kBT l* 3 *ρ*3 0 *ρ*2 <sup>0</sup>*<sup>ρ</sup>* <sup>þ</sup> *<sup>ρ</sup>*0*ρ*<sup>2</sup> <sup>þ</sup> *<sup>ρ</sup>*<sup>3</sup> � � 9 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>= >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>;

in which <sup>∇</sup><sup>2</sup> <sup>¼</sup> *<sup>∂</sup>*<sup>2</sup>

**Author details**

P.R. China

**11**

Hui Cheng<sup>1</sup> and Tian-You Fan<sup>2</sup>

*<sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> *<sup>∂</sup>*<sup>2</sup>

*Probe on Rupture Theory of Soft-Matter Quasicrystals DOI: http://dx.doi.org/10.5772/intechopen.89813*

to variation of mass density, respectively.

valid only for the 10-fold symmetry quasicrystals.

\*

\*Address all correspondence to: tyfan2013@163.com

provided the original work is properly cited.

1 School of Mathematics and Physics, Hebei University of Engineering, Handan,

© 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,

2 School of Physics, Beijing Institute of Technology, Beijing, P.R. China

*<sup>∂</sup>y*<sup>2</sup> <sup>þ</sup> *<sup>∂</sup>*<sup>2</sup>

*<sup>∂</sup>z*<sup>2</sup> , ∇<sup>2</sup>

<sup>1</sup> <sup>¼</sup> *<sup>∂</sup>*<sup>2</sup> *<sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> *<sup>∂</sup>*<sup>2</sup>

**j***Vy* þ **k***Vz*, **u** ¼ **i***ux* þ **j***uy* þ **k***uz*, and *C*11,*C*12,*C*13,*C*33,*C*44,*C*<sup>66</sup> ¼ ð Þ *C*<sup>11</sup> � *C*<sup>12</sup> *=*2 are the phonon elastic constants, *K*1,*K*2, *K*3,*K*<sup>4</sup> are the phason elastic constants, *R* is the phonon-phason coupling constant, *η* is the fluid dynamic viscosity, Γ*<sup>u</sup>* and Γ*<sup>w</sup>* the phonon and phason dissipation coefficients, and *A* and *B* the material constants due

Equation (A1) is the final governing equations of dynamics of soft-matter quasicrystals of 10-fold symmetry in three-dimensional case with field variables *ux*, *uy*, *uz*, *wx*, *wy*,*Vx*,*Vy*,*Vz*, *ρ* and *p*; the amount of the field variables is 10, and the amount of field equations is 10 too; among them are: (A1a) (the first of (A1)) is the mass conservation equation, (A1b)–(A1d) (the second to fourth of (A1)) are the momentum conservation equations or generalized Navier-Stokes equations, (A1e)–(A1g) (the fifth to seventh of (A1)) are the equations of motion of phonons due to the symmetry breaking, (A1h) and (A1i) (the eighth to ninth of (A1)) are the phason dissipation equations, and (A1j) (the tenth of (A1)) is the equation of state, respectively. The equations are consistent to be mathematical solvability, if there is lack of the equation of state, the equation system is not closed, and has no meaning mathematically and physically. This shows that the equation of state is necessary. The equation set (A1) is the three-dimensional form of the equations, and in our solution, we computed only the two-dimensional form, i.e., the plane field form; in this special case, the 5- and 10-fold symmetry quasicrystals have the same governing equations, but in the three-dimensional case, the equation set (A1) is

*<sup>∂</sup>y*<sup>2</sup> , <sup>∇</sup> <sup>¼</sup> **<sup>i</sup>** *<sup>∂</sup>*

*<sup>∂</sup><sup>x</sup>* <sup>þ</sup> **<sup>j</sup>** *<sup>∂</sup>*

*<sup>∂</sup><sup>y</sup>* <sup>þ</sup> **<sup>k</sup>** *<sup>∂</sup>*

*<sup>∂</sup><sup>z</sup>* , **V** ¼ **i***Vx* þ

**A. Appendix**

*∂ρ ∂t*

*<sup>∂</sup>*ð Þ *<sup>ρ</sup>Vx ∂t* þ

þ *C*<sup>66</sup>

<sup>þ</sup>*<sup>R</sup> <sup>∂</sup>*

*∂ ρVy* � � *∂t* þ

�*<sup>R</sup> <sup>∂</sup> ∂x*

*<sup>∂</sup>*ð Þ *<sup>ρ</sup>Vz ∂t* þ

þ *C*<sup>44</sup>

¼ *Vx* þ Γ**<sup>u</sup>**

þð Þ *C*<sup>13</sup> þ *C*<sup>44</sup>

¼ *Vy* þ Γ**<sup>u</sup>**

þð Þ *C*<sup>13</sup> þ *C*<sup>44</sup>

*∂*2 *<sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> *<sup>C</sup>*<sup>44</sup>

*∂ux ∂t* þ *Vx ∂ux <sup>∂</sup><sup>x</sup>* <sup>þ</sup> *Vy*

*∂uy ∂t* þ *Vx ∂uy <sup>∂</sup><sup>x</sup>* <sup>þ</sup> *Vy*

*∂uz ∂t* þ *Vx ∂uz <sup>∂</sup><sup>x</sup>* <sup>þ</sup> *Vy*

þ *C*<sup>44</sup>

<sup>¼</sup> <sup>Γ</sup>**<sup>w</sup>** *<sup>K</sup>*1∇<sup>2</sup>

<sup>¼</sup> <sup>Γ</sup>**<sup>w</sup>** *<sup>K</sup>*1∇<sup>2</sup>

*p* ¼ *f*ð Þ¼ *ρ* 3

*∂wx ∂t* þ *Vx ∂wx <sup>∂</sup><sup>x</sup>* <sup>þ</sup> *Vy*

*∂wy ∂t* þ *Vx ∂wy <sup>∂</sup><sup>x</sup>* <sup>þ</sup> *Vy*

**10**

þð Þ *C*<sup>12</sup> þ *C*<sup>66</sup>

*∂wx <sup>∂</sup><sup>y</sup>* � *<sup>∂</sup>wy ∂x* � �

*∂*2 *<sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> *<sup>C</sup>*<sup>44</sup>

> " *C*<sup>11</sup> *∂*2 *<sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> *<sup>C</sup>*<sup>66</sup>

"

*∂*2 *uz ∂x∂z*

þ ∇ � ð Þ¼ *ρ***V** 0

*Fracture Mechanics Applications*

*∂*2 *<sup>∂</sup>y*<sup>2</sup> <sup>þ</sup> *<sup>C</sup>*<sup>44</sup>

� �

*<sup>∂</sup><sup>x</sup>* <sup>∇</sup><sup>1</sup> � **<sup>w</sup>** � *<sup>R</sup> <sup>∂</sup>*

*∂ VxρVy* � � *∂x* þ

> *∂*2 *ux ∂x∂y*

*<sup>∂</sup>*ð Þ *VxρVz ∂x* þ

*∂*2

*∂ux ∂y* þ *Vz ∂ux ∂z*

<sup>þ</sup> *<sup>R</sup> <sup>∂</sup>*

� *<sup>R</sup> <sup>∂</sup> ∂x*

*∂uz ∂y* þ *Vz ∂uz*

*∂*2 *<sup>∂</sup>y*<sup>2</sup> <sup>þ</sup> *<sup>C</sup>*<sup>33</sup>

> *∂wx ∂y* þ *Vz ∂wx ∂z*

*∂*2 *wx <sup>∂</sup>z*<sup>2</sup> <sup>þ</sup> *<sup>K</sup>*<sup>2</sup>

*∂wy ∂y Vz ∂wy ∂z*

*∂*2 *wy <sup>∂</sup>z*<sup>2</sup> <sup>þ</sup> *<sup>R</sup> <sup>∂</sup> ∂x ∂ux ∂y* þ *∂uy ∂x* � �

<sup>0</sup>*<sup>ρ</sup>* <sup>þ</sup> *<sup>ρ</sup>*0*ρ*<sup>2</sup> <sup>þ</sup> *<sup>ρ</sup>*<sup>3</sup> � �

� �

<sup>1</sup>*wx* þ *K*<sup>4</sup>

<sup>1</sup>*wy* þ *K*<sup>4</sup>

*kBT l* 3 *ρ*3 0 *ρ*2

*∂*2 *ux ∂x∂y*

*∂uy ∂y* þ *Vz ∂uy ∂z*

ð Þ *C*<sup>12</sup> þ *C*<sup>66</sup>

*∂*2 *uz ∂y∂z*

*<sup>∂</sup>*ð Þ *VxρVx ∂x* þ

> *∂*2 *∂z*<sup>2</sup>

*∂y ∂wx <sup>∂</sup><sup>y</sup>* � *<sup>∂</sup>wy ∂x* � �

*∂ VyρVx* � � *∂y*

*∂ VyρVy* � � *∂y*

> *∂*2 *<sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> *<sup>C</sup>*<sup>11</sup>

þ *C*<sup>66</sup>

� *<sup>R</sup> <sup>∂</sup> ∂y*

� �

*∂ VyρVz* � � *∂y*

þ

þ

þ

*<sup>∂</sup>y*<sup>2</sup> <sup>þ</sup> ð Þ *<sup>C</sup>*<sup>33</sup> � *<sup>C</sup>*<sup>13</sup> � *<sup>C</sup>*<sup>44</sup>

*∂*2 *<sup>∂</sup>y*<sup>2</sup> <sup>þ</sup> *<sup>C</sup>*<sup>44</sup>

*<sup>∂</sup><sup>x</sup>* <sup>∇</sup><sup>1</sup> � **<sup>w</sup>** � *<sup>R</sup> <sup>∂</sup>*

þ *C*<sup>66</sup>

*<sup>∂</sup><sup>z</sup>* <sup>¼</sup> *Vz* <sup>þ</sup> <sup>Γ</sup>**<sup>u</sup>**

*uz* #

*∂ ∂y ∂ <sup>∂</sup><sup>x</sup>* � *<sup>∂</sup> ∂y* � �

� � � �

*∂wx <sup>∂</sup><sup>y</sup>* � *<sup>∂</sup>wy ∂x* � �

> *∂*2 *∂z*<sup>2</sup>

� �

*ux* þ ð Þ *C*<sup>12</sup> þ *C*<sup>66</sup>

*<sup>∂</sup>*ð Þ *VzρVx <sup>∂</sup><sup>z</sup>* ¼ � *<sup>∂</sup><sup>p</sup>*

> *∂*2 *uy ∂x∂y*

� ð Þ *<sup>A</sup>* � *<sup>B</sup>* <sup>1</sup> *ρ*0 *∂δρ ∂x*

*∂ VzρVy* � � *<sup>∂</sup><sup>z</sup>* ¼ � *<sup>∂</sup><sup>p</sup> ∂y* <sup>þ</sup> *<sup>η</sup>*∇<sup>2</sup> *Vy* þ 1 3 *η ∂ ∂y* ∇ � **V**

> *∂*2 *∂z*<sup>2</sup>

*ρ*0 *∂δρ ∂y*

*∂*2 *∂z*<sup>2</sup>

*∂*2 *<sup>∂</sup>y*<sup>2</sup> <sup>þ</sup> *<sup>C</sup>*<sup>44</sup>

<sup>∇</sup><sup>1</sup> � **<sup>w</sup>** � ð Þ *<sup>A</sup>* � *<sup>B</sup>* <sup>1</sup>

*<sup>∂</sup>*ð Þ *VzρVz <sup>∂</sup><sup>z</sup>* ¼ � *<sup>∂</sup><sup>p</sup>*

> *∂*2 *∂z*<sup>2</sup>

*∂y ∂wx <sup>∂</sup><sup>y</sup>* � *<sup>∂</sup>wy ∂x* � �#

*∂*2 *<sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> *<sup>C</sup>*<sup>11</sup>

� �

**A.1 The final governing equations of generalized dynamics of soft-matter**

In the text, Eq. (2) is the planar field form only. In book [24], it provides an explicit form of the equations and lists below soft-matter quasicrystals with 10-fold symmetry:

*<sup>∂</sup><sup>x</sup>* <sup>þ</sup> *<sup>η</sup>*∇<sup>2</sup>

þ ð Þ *C*<sup>13</sup> þ *C*<sup>44</sup> � *C*<sup>11</sup>

*<sup>∂</sup><sup>z</sup>* <sup>þ</sup> *<sup>η</sup>*∇<sup>2</sup>

*ux* þ ð Þ *C*<sup>12</sup> þ *C*<sup>66</sup>

*∂*2 *<sup>∂</sup>y*<sup>2</sup> <sup>þ</sup> *<sup>C</sup>*<sup>44</sup>

ð Þ *C*<sup>13</sup> þ *C*<sup>44</sup>

*wy* <sup>þ</sup> *<sup>R</sup> <sup>∂</sup> ∂x ∂ux <sup>∂</sup><sup>x</sup>* � *<sup>∂</sup>uy ∂y* � �

<sup>þ</sup> *<sup>R</sup> <sup>∂</sup> ∂y ∂ux <sup>∂</sup><sup>x</sup>* � *<sup>∂</sup>uy ∂y*

� � ��

� �

� *<sup>R</sup> <sup>∂</sup> ∂y* ∇<sup>1</sup> � **w** #

"

*Vx* þ 1 3 *η ∂ <sup>∂</sup><sup>x</sup>* <sup>∇</sup> � **<sup>V</sup>**

*uy* þ ð Þ *C*<sup>13</sup> þ *C*<sup>44</sup>

*Vz* þ 1 3 *η ∂ ∂z* ∇ � **V**

*∂*2 *uy ∂x∂y*

*∂*2 *∂z*<sup>2</sup>

*∂*2 *ux ∂x∂z* þ *∂*2 *uy ∂y∂z* � �

*uy*

� *<sup>R</sup> <sup>∂</sup> ∂y ∂ux ∂y* þ *∂uy ∂x*

*uz* <sup>þ</sup> ð Þ *<sup>C</sup>*<sup>13</sup> <sup>þ</sup> *<sup>C</sup>*<sup>44</sup> � <sup>B</sup> *<sup>∂</sup>*

*∂*2 *uz ∂x∂z*

> *∂*2 *uz ∂y∂z*

*∂z*

<sup>þ</sup> ð Þ *<sup>C</sup>*<sup>11</sup> � *<sup>B</sup> <sup>∂</sup>*

<sup>þ</sup> ð Þ *<sup>C</sup>*<sup>11</sup> � *<sup>B</sup> <sup>∂</sup>*

<sup>∇</sup> � **<sup>u</sup>** � ð Þ *<sup>A</sup>* � *<sup>B</sup>* <sup>1</sup>

*<sup>∂</sup><sup>x</sup>* <sup>∇</sup> � **<sup>u</sup>**

9

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>=

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>;

(A1)

*∂y* ∇ � **u**

> *ρ*0 *∂δρ ∂z*

**quasicrystals with 10-fold symmetry in three dimension**

in which <sup>∇</sup><sup>2</sup> <sup>¼</sup> *<sup>∂</sup>*<sup>2</sup> *<sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> *<sup>∂</sup>*<sup>2</sup> *<sup>∂</sup>y*<sup>2</sup> <sup>þ</sup> *<sup>∂</sup>*<sup>2</sup> *<sup>∂</sup>z*<sup>2</sup> , ∇<sup>2</sup> <sup>1</sup> <sup>¼</sup> *<sup>∂</sup>*<sup>2</sup> *<sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> *<sup>∂</sup>*<sup>2</sup> *<sup>∂</sup>y*<sup>2</sup> , <sup>∇</sup> <sup>¼</sup> **<sup>i</sup>** *<sup>∂</sup> <sup>∂</sup><sup>x</sup>* <sup>þ</sup> **<sup>j</sup>** *<sup>∂</sup> <sup>∂</sup><sup>y</sup>* <sup>þ</sup> **<sup>k</sup>** *<sup>∂</sup> <sup>∂</sup><sup>z</sup>* , **V** ¼ **i***Vx* þ **j***Vy* þ **k***Vz*, **u** ¼ **i***ux* þ **j***uy* þ **k***uz*, and *C*11,*C*12,*C*13,*C*33,*C*44,*C*<sup>66</sup> ¼ ð Þ *C*<sup>11</sup> � *C*<sup>12</sup> *=*2 are the phonon elastic constants, *K*1,*K*2, *K*3,*K*<sup>4</sup> are the phason elastic constants, *R* is the phonon-phason coupling constant, *η* is the fluid dynamic viscosity, Γ*<sup>u</sup>* and Γ*<sup>w</sup>* the phonon and phason dissipation coefficients, and *A* and *B* the material constants due to variation of mass density, respectively.

Equation (A1) is the final governing equations of dynamics of soft-matter quasicrystals of 10-fold symmetry in three-dimensional case with field variables *ux*, *uy*, *uz*, *wx*, *wy*,*Vx*,*Vy*,*Vz*, *ρ* and *p*; the amount of the field variables is 10, and the amount of field equations is 10 too; among them are: (A1a) (the first of (A1)) is the mass conservation equation, (A1b)–(A1d) (the second to fourth of (A1)) are the momentum conservation equations or generalized Navier-Stokes equations, (A1e)–(A1g) (the fifth to seventh of (A1)) are the equations of motion of phonons due to the symmetry breaking, (A1h) and (A1i) (the eighth to ninth of (A1)) are the phason dissipation equations, and (A1j) (the tenth of (A1)) is the equation of state, respectively. The equations are consistent to be mathematical solvability, if there is lack of the equation of state, the equation system is not closed, and has no meaning mathematically and physically. This shows that the equation of state is necessary.

The equation set (A1) is the three-dimensional form of the equations, and in our solution, we computed only the two-dimensional form, i.e., the plane field form; in this special case, the 5- and 10-fold symmetry quasicrystals have the same governing equations, but in the three-dimensional case, the equation set (A1) is valid only for the 10-fold symmetry quasicrystals.

## **Author details**

Hui Cheng<sup>1</sup> and Tian-You Fan<sup>2</sup> \*

1 School of Mathematics and Physics, Hebei University of Engineering, Handan, P.R. China

2 School of Physics, Beijing Institute of Technology, Beijing, P.R. China

\*Address all correspondence to: tyfan2013@163.com

© 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.

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[3] Hayashida K, Dotera T, Takano A, Matsushita Y. Polymeric quasicrystal: Mesoscopic quasicrystalline tiling in ABC star polymers. Physical Review Letters. 2007;**98**(19):195502. DOI: 10.1103/PhysRevLett.98.195502

[4] Talapin DV, Shevchenko EV, Bodnarchuk MI, Ye X, Chen J, Murray CB. Quasicrystalline order in self-assembled binary nanoparticle superlattices. Nature. 2009;**461**(7266): 964-967. DOI: 10.1038/nature08439

[5] Fischer S, Exner A, Zielske K, Perlich J, Deloudi S, Steurer W, et al. Colloidal quasicrystals with 12-fold and 18-fold diffraction symmetry. Proceedings of the National Academy of Sciences. 2011;**108**(5):1810-1814. DOI: 10.1073/pnas.1008695108

[6] Cheng SZD. Giant surfactants based on precisely functionalized POSS nanoatoms: Tuning from crystals to Frank-Kasper phases and quasicrystals. In: First Annual Symposium on Frontiers of Soft Matter Science and Engineering, Beijing; 2015. Bulletin of the American Physical Society, 2016

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Mathematical Models and Solutions. Beijing/Heidelberg: Beijing Institute of Technology Press/Springer-Verlag; 2017

[25] Landau LD, Lifshitz EM. Theoretical Physics V, Statistical Physics, Part I. 3rd ed. New York: Pergamon Press; 1980

[26] Anderson PW. Basic Notations of Condensed Matter Physics. Menlo Park:

[28] Fan L, Fan TY. Equation of state of structured liquid. In: First Annual Symposium on Frontiers of Soft Matter Science and Engineering; December 12,

Benjamin Cummings; 1984

2015; Beijing, China; 2015

Press; 1980

[27] Lifshitz EM, Pitaevskii LP. Theoretical Physics V, Statistical Physics, Part II. New York: Pergamon

Soft-Matter Quasicrystals—

[16] Martinez CJ, Lewis JA. Shape evolution and stress development during latex-silica film formation. Langmuir. 2002;**18**(12):4689-4698. DOI:

[17] Tirumkudulu MS, Russel WB. Role of capillary stresses in film formation. Langmuir. 2004;**20**(7):2947-2961. DOI:

[18] Bohn S, Pauchard L, Couder Y. Hierarchical crack pattern as formed by successive domain divisions. Physical Review E. 2005;**71**(4):046214. DOI: 10.1103/PhysRevE.71.046215

[19] Griffith AA. The phenomena of rupture and flow in solids. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering

Sciences. 1921;**221**:582-593. DOI:

[20] Fan TY. Mathematical Theory of Elasticity of Quasicrystals and Its Applications. 1st ed./2nd ed. Beijing/ Heidelberg: Science Press/Springer-Verlag; 2010/2016. DOI: 10.1007/978-3-

[21] Fan TY. Poisson bracket method and its applications to quasicrystals, liquid crystals and a class of soft matter. Acta Mechanica Sinica. 2013;**45**(4):548-559. DOI: 10.6052/0459-1879-12-346 (in

[22] Fan TY. Equation system of generalized hydrodynamics of softmatter quasicrystals. Applied Mathematics and Mechanics. 2016;

hydrodynamics for second 2D softmatter quasicrystals. Applied Mathematics and Mechanics. 2017;

**37**(4):331-344 (in Chinese)

[23] Fan TY. Generalized

**38**(2):189-199 (in Chinese)

10.1021/la0114833

10.1021/la0356250

10.2307/91192

642-14643-5

Chinese)

**13**

[8] Goehring L, Conroy R, Akhter A, et al. Evolution of mud-crack patterns during repeated drying cycles. Soft Matter. 2010;**6**(15):3562-3567. DOI: 10.1039/b922206e

[9] Carbone G, Pierro E, Gorb SN. Origin of the superior adhesive performance of mushroom-shaped microstructured surfaces. Soft Matter. 2011;**7**(12): 5545-5552. DOI: 10.1039/c0sm01482f

[10] Xuan Y, Guo X, Cui Y, et al. Crackfree controlled wrinkling of a bilayer film with a gradient interface. Soft Matter. 2012;**8**(37):9603-9609. DOI: 10.1039/c2sm25487e

[11] Haque MA, Kurokawa T, Kamita G, et al. Lamellar bilayers as reversible sacrificial bonds to toughen hydrogel: Hysteresis, self-recovery, fatigue resistance, and crack blunting. Macromolecules. 2011;**44**(22): 8916-8924. DOI: 10.1021/ma201653t

[12] Hatton B, Mishchenko L, Davis S, et al. Assembly of large-area, highly ordered, crack-free inverse opal films. Proceedings of the National Academy of Sciences. 2010;**107**(23):10354-10359. DOI: 10.1073/pnas.1000954107

[13] Lazarus V, Pauchard L. From craquelures to spiral crack patterns: Influence of layer thickness on the crack patterns induced by desiccation. Soft Matter. 2011;**7**(6):2552-2559. DOI: 10.1039/c0sm00900h

[14] Tirumkudulu MS, Russel WB. Cracking in drying latex films. Langmuir. 2005;**21**(11):4938-4948. DOI: 10.1021/la048298k

[15] van der Kooij HM, Sprakel J. Watching paint dry; more exciting than *Probe on Rupture Theory of Soft-Matter Quasicrystals DOI: http://dx.doi.org/10.5772/intechopen.89813*

it seems. Soft Matter. 2015;**11**(32): 6353-6359. DOI: 10.1039/C5SM01505G

**References**

nature02368

[1] Zeng X, Ungar G, Liu Y, Percec V, Dulcey AE, Hobbs JK. Supramolecular dendritic liquid quasicrystals. Nature. 2004;**428**(6979):157-160. DOI: 10.1038/

*Fracture Mechanics Applications*

National Academy of Sciences. 2016;**113**:

[9] Carbone G, Pierro E, Gorb SN. Origin of the superior adhesive performance of mushroom-shaped microstructured surfaces. Soft Matter. 2011;**7**(12): 5545-5552. DOI: 10.1039/c0sm01482f

[10] Xuan Y, Guo X, Cui Y, et al. Crackfree controlled wrinkling of a bilayer film with a gradient interface. Soft Matter. 2012;**8**(37):9603-9609. DOI:

[11] Haque MA, Kurokawa T, Kamita G, et al. Lamellar bilayers as reversible sacrificial bonds to toughen hydrogel: Hysteresis, self-recovery, fatigue resistance, and crack blunting. Macromolecules. 2011;**44**(22): 8916-8924. DOI: 10.1021/ma201653t

[12] Hatton B, Mishchenko L, Davis S, et al. Assembly of large-area, highly ordered, crack-free inverse opal films. Proceedings of the National Academy of Sciences. 2010;**107**(23):10354-10359. DOI: 10.1073/pnas.1000954107

[13] Lazarus V, Pauchard L. From craquelures to spiral crack patterns: Influence of layer thickness on the crack patterns induced by desiccation. Soft Matter. 2011;**7**(6):2552-2559. DOI:

[14] Tirumkudulu MS, Russel WB. Cracking in drying latex films.

[15] van der Kooij HM, Sprakel J. Watching paint dry; more exciting than

Langmuir. 2005;**21**(11):4938-4948. DOI:

10.1039/c0sm00900h

10.1021/la048298k

[8] Goehring L, Conroy R, Akhter A, et al. Evolution of mud-crack patterns during repeated drying cycles. Soft Matter. 2010;**6**(15):3562-3567. DOI:

1392-1400

10.1039/b922206e

10.1039/c2sm25487e

[2] Takano A, Kawashima W, Noro A, Isono Y, Tanaka N, Dotera T, et al. A mesoscopic Archimedean tiling having a new complexity in an ABC star polymer. Journal of Polymer Science Part B: Polymer Physics. 2005;**43**(18): 2427-2432. DOI: 10.1002/polb.20537

[3] Hayashida K, Dotera T, Takano A, Matsushita Y. Polymeric quasicrystal: Mesoscopic quasicrystalline tiling in ABC star polymers. Physical Review Letters. 2007;**98**(19):195502. DOI: 10.1103/PhysRevLett.98.195502

[4] Talapin DV, Shevchenko EV, Bodnarchuk MI, Ye X, Chen J, Murray CB. Quasicrystalline order in self-assembled binary nanoparticle superlattices. Nature. 2009;**461**(7266): 964-967. DOI: 10.1038/nature08439

[5] Fischer S, Exner A, Zielske K, Perlich J, Deloudi S, Steurer W, et al. Colloidal quasicrystals with 12-fold and

18-fold diffraction symmetry.

10.1073/pnas.1008695108

Physical Society, 2016

**12**

Proceedings of the National Academy of Sciences. 2011;**108**(5):1810-1814. DOI:

[6] Cheng SZD. Giant surfactants based on precisely functionalized POSS nanoatoms: Tuning from crystals to Frank-Kasper phases and quasicrystals. In: First Annual Symposium on Frontiers of Soft Matter Science and Engineering, Beijing; 2015. Bulletin of the American

[7] Yue K, Huang MJ, Marson R, He JL, Huang JH, Zhou Z, et al. Geometry induced sequence of nanoscale Frank-Kasper and quasicrystal mesophases in giant surfactants. Proceedings of the

[16] Martinez CJ, Lewis JA. Shape evolution and stress development during latex-silica film formation. Langmuir. 2002;**18**(12):4689-4698. DOI: 10.1021/la0114833

[17] Tirumkudulu MS, Russel WB. Role of capillary stresses in film formation. Langmuir. 2004;**20**(7):2947-2961. DOI: 10.1021/la0356250

[18] Bohn S, Pauchard L, Couder Y. Hierarchical crack pattern as formed by successive domain divisions. Physical Review E. 2005;**71**(4):046214. DOI: 10.1103/PhysRevE.71.046215

[19] Griffith AA. The phenomena of rupture and flow in solids. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 1921;**221**:582-593. DOI: 10.2307/91192

[20] Fan TY. Mathematical Theory of Elasticity of Quasicrystals and Its Applications. 1st ed./2nd ed. Beijing/ Heidelberg: Science Press/Springer-Verlag; 2010/2016. DOI: 10.1007/978-3- 642-14643-5

[21] Fan TY. Poisson bracket method and its applications to quasicrystals, liquid crystals and a class of soft matter. Acta Mechanica Sinica. 2013;**45**(4):548-559. DOI: 10.6052/0459-1879-12-346 (in Chinese)

[22] Fan TY. Equation system of generalized hydrodynamics of softmatter quasicrystals. Applied Mathematics and Mechanics. 2016; **37**(4):331-344 (in Chinese)

[23] Fan TY. Generalized hydrodynamics for second 2D softmatter quasicrystals. Applied Mathematics and Mechanics. 2017; **38**(2):189-199 (in Chinese)

[24] Fan TY. Generalized Dynamics of Soft-Matter Quasicrystals— Mathematical Models and Solutions. Beijing/Heidelberg: Beijing Institute of Technology Press/Springer-Verlag; 2017

[25] Landau LD, Lifshitz EM. Theoretical Physics V, Statistical Physics, Part I. 3rd ed. New York: Pergamon Press; 1980

[26] Anderson PW. Basic Notations of Condensed Matter Physics. Menlo Park: Benjamin Cummings; 1984

[27] Lifshitz EM, Pitaevskii LP. Theoretical Physics V, Statistical Physics, Part II. New York: Pergamon Press; 1980

[28] Fan L, Fan TY. Equation of state of structured liquid. In: First Annual Symposium on Frontiers of Soft Matter Science and Engineering; December 12, 2015; Beijing, China; 2015

**Chapter 2**

**Abstract**

**1. Introduction**

**15**

Application of J Integral for the

Fracture Assessment of Welded

*Zoltan Major, Daniel Kimpfbeck and Matei C. Miron*

For many demanding applications of engineering plastics, fracture behaviour under various loading conditions is of prime practical importance. It is well known that fracture properties of plastics are significantly affected by the loading rate, temperature and both local and global stress states. The limitations associated with conventional fracture test methods may, at least in principle, be overcome by the use of appropriate fracture mechanical approaches, which properly account for the temperature and rate dependence of the mechanical behaviour of plastics and should provide geometry-independent fracture toughness values. To provide an additional contribution to this application, fracture tests were performed on both 15- and 20-mm-thick bulk-extruded sheets of a polypropylene random copolymer (PP(RC)) and on four different configurations of their welded joints. The fully ductile fracture range was determined by rate-dependent tests on single CT specimens, and fracture toughness values were derived at the peak loads (*JFmax* and *CTODFmax*). Fracture toughness values were determined for stable crack extension based on the *J-Δa* and/or *CTOD-Δa R*-curves using single and multiple specimens in terms of various definitions of the crack initiation (*J0.2, J0.2BL* or *δ0.2*) toughness values. As expected, both methods revealed distinct differences between the bulk materials and the welded joints. These differences were found to depend on the loading rate, the weld configuration and on

**Keywords:** bulk polymer, welded polymeric structures, elastic–plastic fracture

For many demanding applications of engineering plastics, fracture behaviour under various loading conditions is of prime practical importance. In this context it is well known that fracture properties of plastics are significantly affected by the loading rate, temperature and both local (if notches or cracks are present) and global stress states (component geometry or specimen configuration). As a result of the complex effects of these parameters, fracture values determined by conventional test methods (e.g. unnotched and notched Charpy fracture energies) are only of very limited use for material characterisation, especially for engineering design purposes. The limitations associated with conventional testing methods may be overcome by the use of appropriate fracture mechanical approaches, which properly account for temperature and rate dependence of the mechanical behaviour of

mechanics (EPFM), CTOD, J integral, ductile-brittle transition

Polymeric Components

the data reduction method (*J* integral or *CTOD*).

## **Chapter 2**
