Indentation on one-dimensional hexagonal quasicrystals: general theory and complete exact solutions

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Indentation on one-dimensionalhexagonal quasicrystals: general theoryand complete exact solutionsY.F. Wu a , W.Q. Chen b & X.Y. Li ca Department of Civil Engineering, Zhejiang University, ZijingangCampus, Hangzhou 310058, Chinab Department of Engineering Mechanics, Zhejiang University,Yuquan Campus, Hangzhou 310027, Chinac School of Mechanics and Engineering, Southwest JiaotongUniversity, Chengdu 610031, ChinaVersion of record first published: 26 Oct 2012.

To cite this article: Y.F. Wu , W.Q. Chen & X.Y. Li (2013): Indentation on one-dimensional hexagonalquasicrystals: general theory and complete exact solutions, Philosophical Magazine, 93:8, 858-882

To link to this article: http://dx.doi.org/10.1080/14786435.2012.735772

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Indentation on one-dimensional hexagonal quasicrystals: general

theory and complete exact solutions

Y.F. Wua, W.Q. Chenb and X.Y. Lic*

aDepartment of Civil Engineering, Zhejiang University, Zijingang Campus, Hangzhou310058, China; bDepartment of Engineering Mechanics, Zhejiang University, Yuquan

Campus, Hangzhou 310027, China; cSchool of Mechanics and Engineering,Southwest Jiaotong University, Chengdu 610031, China

(Received 29 October 2011; final version received 27 September 2012)

This paper presents a general account of the indentation responses of aone-dimensional hexagonal quasicrystal half-space pressed by an axisym-metric rigid punch. Based on Green’s functions of the half-space subjectedto point sources on the surface, the mixed boundary value problem istransformed to integral equations and solved exactly using the results of thepotential theory method. Explicit expressions for the generalised pressuresand indentation forces are derived for three common indenters (cylinder,cone and approximate sphere) in a systematic manner. For conical andspherical indenters, relations between the contact radius and indentationloads are determined. The coupling phonon–phason fields in the half-spaceunder indentation are accurately expressed in terms of elementaryfunctions. Numerical calculations are performed and discussions on relatedphysical phenomena are given. The present exact solutions can serve asbenchmarks for approximate or numerical analyses and can guide theexperimental characterisation of material properties of quasicrystals.

Keywords: quasicrystals; half-space; Green’s functions; indentation; exactsolutions

1. Introduction

The quasicrystal (QC) is a new structural form of solid materials. Since its discoveryby Shechtman et al. [1], the study of the mechanical behavior of QCs from theviewpoint of linear elasticity has attracted a lot of research interest [2–6]. Amongthem, significant achievements have been made on dislocation and crack problems,which are of practical importance because QCs are very brittle at room and evenelevated temperatures [7–10].

A one-dimensional (1D) QC is defined as a three-dimensional (3D) body which isperiodic on a plane composed of two directions but quasi-periodic in the thirddirection. Three-dimensional analyses of 1D QCs have been made. For example,Wang et al. [11] derived all the possible point and space groups of 1D QCs. Li andFan [12] and Fan et al. [13] investigated some dislocation problems of 1D QCs.

*Corresponding author. Email: [email protected]

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By noticing the mathematical similarity with the theory of transversely isotropicpiezoelectric materials [14,15], Chen et al. [16] presented a general solution for3D elastic problems of 1D hexagonal QCs, with which the fundamental solutions(or Green’s functions) for point phonon or phason sources are also derived.

The generalised elasticity of QCs is described in terms of the two types of elasticfields, i.e. the phonon (conventional) elastic field and the phason field. Phason degreeof freedom arises as a consequence of the quasi-periodic symmetric of QCs. Phasondisplacement describes the atomic jump from one position to another one nearbyhaving a similar local environment or rearrangement of atoms from one potentialvalley to another, which will break the quasi-periodic symmetric property of QCsand change the system elastic energy.

Considerable research efforts have been devoted to theoretical calculations ofelastic constants of QCs. The methods include Monte Carlo simulation [17–19],relaxation simulation [20,21] and transfer-matrix method [22]. The phason elasticconstant and phason–phonon coupling constant were recently determined experi-mentally by measuring the diffuse scattering due to phason disorder. de Boissieuet al. [23] and Letoublon et al. [24] first determined the phason elastic constant ratioand absolute value of the Al–Pd–Mn icosahedral phase from measurements ofdiffuse scattering located close to Bragg reflections. Edagawa [25] first evaluated thephason–phonon coupling constant by X-ray diffraction measurements of phononstrain introduced at the phase transition.

Recently, indentation techniques and atomic force microscopy (AFM) have beenused to study the mechanical behavior of QCs [26–32]. Most of the experimentalstudies are qualitative and based on conventional techniques developed for elasticmaterials indentation testing [33]. The coupling between the phonon and phasonfields, however, should be carefully identified so as to ascertain to what extent it willaffect the indentation responses of QCs. This can only be done based on the fullycoupled elasticity theory of QCs [11]. Such contact analysis has been carried outtheoretically by Peng and Fan [34], who derived an integral-form solution for acylindrical punch indenting a 1D hexagonal quasicrystal half-space by the Fourierseries expansion and Hankel transform. Zhou et al. [35] examined the axisymmetriccontact problems of cubic QCs in a similar way, and presented two general theoremsregarding the distribution of generalised pressures and displacements underneath therigid punch. It should be pointed out that the relation between the indentation loadand the indentation depth, which is very useful in experimental studies usingindentation techniques, has not been presented in the literature. Furthermore,although the cone (which is usually applied to model the Berkovich and Vickersindenter) and the sphere are the most commonly used shapes in the theoreticalanalysis of indentation experiments, contact solutions of QCs corresponding toconical and spherical indenters have never been reported.

Inspired by the work of Chen et al. [36], this paper aims at seeking exact solutionsof the contact problems of 1D hexagonal QCs pressed by three common types ofindenters, and presenting the explicit relations between the indentation loadand indentation depth or contact radius. The half-space Green’s functions arederived by analogy with those for transversely isotropic piezoelectric materials,which were first obtained by Ding et al. [15] and later documented in the book ofDing and Chen [37]. Then, the boundary integral equations are obtained using the

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superposition principle. Exact and complete fundamental solutions corresponding tothe indenters under consideration are systematically presented in terms of elementaryfunctions. Section 5 presents some numerical results and the discussions. Finally,concluding remarks are made in Section 6.

2. Governing equations and half-space Green’s functions

In the cylindrical coordinate system ðr, �, zÞ, the generalised Hooke’s law of the 1Dhexagonal QCs, which is periodic in the r� � plane and quasi-periodic along thez-direction reads [11]

�rr ¼ c11"rr þ c12"�� þ c13"zz þ R1wzz,

�� ¼ c12"rr þ c11"�� þ c13"zz þ R1wzz,

�zz ¼ c13"rr þ c13"�� þ c33"zz þ R2wzz,

��z ¼ �z� ¼ 2c44"�z þ R3wz�,

�rz ¼ �zr ¼ 2c44"rz þ R3wzr,

�r� ¼ ��r ¼ 2c66"r� ¼ c11 � c12ð Þ"r�,

ð1aÞ

Hzr ¼ 2R3"rz þ K2wzr,

Hz� ¼ 2R3"�z þ K2wz�,

Hzz ¼ R1"rr þ R1"�� þ R2"zz þ K1wzz,

ð1bÞ

where �ij and "ij are the phonon stresses and strains, Hij and wij the phason stressesand strains, cij and Ki are the elastic constants in the phonon and phason fields,respectively, and Ri the phonon–phason coupling constants. The phonon andphason strains can be determined from the generalised displacements ui (phononfield) and wz (phason field) as

"rr ¼@ur@r

, "�� ¼1

r

@u�@�þurr, "zz ¼

@uz@z

, "�z ¼ "z� ¼1

2

@u�@zþ@uz@�

� �,

"rz ¼ "zr ¼1

2

@uz@rþ@ur@z

� �, "r� ¼ "�r ¼

1

2

1

r

@ur@�þ@u�@r�u�r

� �,

ð2aÞ

wzr ¼@wz

@r, wz� ¼

1

r

@wz

@�, wzz ¼

@wz

@z: ð2bÞ

The generalised equilibrium equations in the absence of body forces are

@�rr@rþ@�r�r@�þ@�rz@zþ�rr � ��

r¼ 0,

@�r�@rþ@��r@�þ@��z@zþ2�r�r¼ 0,

@�zr@rþ@��zr@�þ@�zz@zþ�zrr¼ 0,

@Hzr

@rþ@Hz�

r@�þ@Hzz

@zþHzr

r¼ 0:

ð3Þ

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Consider a vertical phonon field force p and phason field force s applied at theorigin located on the surface of the 1D QC half-space, which is parallel with the r� �plane. The problem is axisymmetric with u� ¼ 0 and all the physical variables areindependent of the angle variable �. Noticing the mathematical similarity with theformulations of transversely isotropic piezoelectric materials [15], we can easilyobtain the point-source Green’s functions for QCs as follows:

ur r, zð Þ ¼ �X3i¼1

Air

RiR�i

, wk r, zð Þ ¼X3i¼1

�ikAi1

Ri,

�zl r, zð Þ ¼ �X3i¼1

�ilAizi

R3i

, �zk r, zð Þ ¼ �X3i¼1

�iksiAir

R3i

,

�2 r, zð Þ ¼ 2c66X3i¼1

Air2

R2i R�i

1

Riþ

1

R�i

� �,

ð4Þ

where k ¼ 1, 2 and l ¼ 1, 2, 3, Ri ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ z2i

p, R�i ¼ Ri þ zi, zj ¼ sjz. In the above

equation, siði ¼ 1, 2, 3Þ are the three eigenvalues (with positive real parts) of thecharacteristic equation given in the Appendix 1. Throughout this paper, theeigenvalues are assumed to be distinct. The cases for equal eigenvalues are notconsidered for simplicity, and the corresponding analyses can follow the derivationby Chen [38] for piezoelectric materials. Note that the following notations have beenintroduced to make the Green’s functions in Equation (4) more compact:

w1 ¼ uz, w2 ¼ wz,

�z1 ¼ �zz, �z2 ¼ Hzz, �z3 ¼ �rr þ ��,

�2 ¼ �rr � ��, �z1 ¼ �rz, �z2 ¼ Hzr:

ð5Þ

The material constants appearing in Equation (4) are given in the Appendix 2.The coefficients Ai, which are determined from the boundary conditions at z ¼ 0 aswell as the ‘‘force’’ equilibrium condition [15], turn out to be Ai ¼

P2j¼1 Iijpj, where

I1jI2jI3j

8<:

9=; ¼ 1

2�

�11s1 �21s2 �31s3�11 �21 �31�12 �22 �32

24

35�1

0�1j�2j

8<:

9=;, ð6Þ

with p1 ¼ p, p2 ¼ s, and �ij being the Kronecker delta. As one can see from Equation(4), the Green’s functions for point sources applied on the boundary of a half-spaceare expressed exactly and explicitly in terms of elementary functions. This will be ofgreat convenience and value in the subsequent analysis of contact problems.

3. Boundary conditions and integral equations

A smooth rigid punch is considered to be pressed against the QC half-space z � 0,shown in Figure 1. In such a situation, the field quantities should vanish at infinity,which has been automatically satisfied by the Green’s functions given inEquation (4). The frictionless contact between the rigid indenter and the QChalf-space will be discussed in this paper. Only the displacement in the normal


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direction caused by the penetration of the indenter will be considered, the lateral

displacement is ignored because the ratio between the indentation depth and contact

radius is small. However, when the indenter becomes sharper, the discrepancy

between the Hertz and Sneddon’s classical solution and the solution considering the

lateral displacement will be clearer. For the discussions on the effect of lateral

displacement on accurate estimation of the indentation data, one can refer to [39,40].

Thus, according the assumption in this paper, the boundary conditions on the

surface z ¼ 0 can be written as:

0 � r � a : uz r, 0ð Þ ¼ d� f rð Þ

0 � r � a : wz r, 0ð Þ ¼ ’0,

r4 a : �z ¼ 0, Hzz ¼ 0,

r � 0 : �rz ¼ 0,

ð7Þ

where a is the radius of the circular contact area, d is the indentation depth, ’0 the

constant phason field displacement, and the function f ðrÞ is prescribed by the fact

that, when taking the tip as the origin of the coordinates, the indenter profile satisfies

the equation z ¼ f ðrÞ. For a flat-ended cylindrical indenter, fðrÞ ¼ 0. For a conical

indenter fðrÞ ¼ r cot, in which is the half apex angle of the cone. For a spherical

indenter fðrÞ ¼ R�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 � r2p

is approximated by the first item of its corresponding

Taylor series resulting in fðrÞ ¼ r2=ð2RÞ for a� R in the indentation test, in which R

is the radius of the sphere. Thus, ‘sphere’ actually stands for a parabolic shape or the

parabolic approximation of a sphere throughout this paper.Peng and Fan [34] assumed the contact boundary condition of the phason

displacement wzðr, 0Þ ¼ 0; however, we set wzðr, 0Þ ¼ ’0 in this paper, in which ’0 canbe zero or a constant. Since there is no experimental study on whether pressing the

indenter into QCs can result in a phason displacement distribution inside the contact

region or how to apply the phason displacement on the QC surface in the

experiment, it seems that the condition wz(r,0)¼ 0 would be more reasonable.

To show the responses under the phason displacement distribution from the

z

P or S

Quasicrystal half-space

r(0, 0)

Spherical indenter

Figure 1. Spherical indenter pressed into a quasicrystal half-space.

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perspective of theoretical study, we assume the indenter can induce not only the

phonon filed displacement but also the phason filed displacement, and the applied

phason filed displacement is equivalent inside the contact region.By the superposition principle, the generalised indentation displacements can be

expressed in terms of the generalised pressures by the following integrals over the

contact area ð0 � r � aÞ

wk r, �, 0ð Þ ¼X2j¼1

kj

Z 2�

0

Z a

0

pj �, �0ð Þ

R0�d�d�0, k ¼ 1, 2ð Þ, ð8Þ

where kj ¼ jk ¼P3

i¼1 �ikIij, R0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ �2 � 2r� cosð� � �0Þ

pis the distance between

two surface points ðr, �0, 0Þ and ð�, �0, 0Þ. Furthermore, the generalised pressures

pk r, �ð Þ ¼ ��zk r, �, 0ð Þ, k ¼ 1, 2ð Þ ð9Þ

are functions to be determined in this study.For axisymmetric problems, both wk and pk (k ¼ 1, 2) are independent of the

angle �. However, we still keep this variable in Equation (8) so that the governing

equations have the same structure as that in Fabrikant [41,42], whose results will be

employed later.

4. Fundamental results of indentation on a hexagonal quasicrystal half-space

From Equation (8), we can obtain

Z 2�

0

Z a

0

pj �, �0ð Þ

R0�d�d�0 ¼

1

�

X2k¼1

�kjwk r, �, 0ð Þ, j ¼ 1, 2ð Þ, ð10Þ

where � ¼ jkjj is the determinant, and �kj are the corresponding cofactors.The integral equations in (10) can be solved either using the property of the Abel

operator or directly using the results in potential theory. Here, we choose the latter

path and the results in Fabrikant [41] and Hanson [43,44] are adopted. According to

Fabrikant [41], Equation (10) can be rewritten as

4

Z r

0

dx

r2 � x2ð Þ1=2

Z a

x

�d�

�2 � x2ð Þ1=2

Lx2

r�

� �pj �, �ð Þ ¼ qj r, �ð Þ, j ¼ 1, 2,ð Þ ð11Þ

where qj ¼1�

P2k¼1

�kjwk r, �, 0ð Þ, and

L kð Þ f �ð Þ ¼1

2�

Z 2�

0

k, � � �0ð Þf �0ð Þd�0, ð12Þ

is the Poisson operator with the kernel

k, �ð Þ ¼1� k2

1þ k2 � 2k cos �: ð13Þ


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Then, the solution to Equation (11) can be obtained by inverting two Abeloperators and one Poisson operator as

pj r, �ð Þ ¼1

�2�j a, r, �ð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 � r2p �

Z a

r

dtffiffiffiffiffiffiffiffiffiffiffiffiffiffit2 � r2p

d

dt�j t, r, �ð Þ

� �, ð14Þ

with

�j ðt, r, �Þ ¼1

t

Z t

0

xdxffiffiffiffiffiffiffiffiffiffiffiffiffiffit2 � x2p

d

dxxL

xr

t2

� �qj ðx, �Þ

h i: ð15Þ

It is seen that the first term on the right-hand side of Equation (14) is singular atr ¼ a for a non-zero �j ða, a, �Þ. For the three indenters in question, we can obtain

�j ðt, r, �Þ ¼

q0j, for flat,

q0j ��t2�1j� cot , for cone,

q0j ��1j�

t2

R , for sphere,

8><>: ð16Þ

with q0j ¼1�

P2k¼1 ð�1jdþ �2j’0Þ.

For a smooth indenter (e.g. conical or spherical indenter), an additionalcondition is required to determine the contact radius a before obtaining completesolutions. This condition is usual to eliminate the stress singularity at r ¼ a.For multi-field coupling materials, e.g. piezoelectric material, the contact radius is

determined by removing the stress singularity; however, there is a singularity in theelectrical field [36,45,46]. For quasicrystals, the requirement for the absence of stresssingularity in the phonon field at r ¼ a will be used to determine the contact radius.It can be seen that the singularity of phason field stress will also disappear when’0 ¼ 0. However, the phason field stress is singular at r ¼ a when ’0 6¼ 0 because weassume the surface phason field displacement ’0 is a constant inside the contact areaand the discontinuity of the phason load at r ¼ a will produce stress singularity atr ¼ a definitely.

From Equations (15) and (16), the condition of removing the phonon stresssingularity leads to the following relations:

q01 ��a

2

�11�

cot ¼ 0, ð17Þ

for the conical indenter, and

q01 ��11�

a2

R¼ 0, ð18Þ

for the spherical indenter. The relations in Equations (17) and (18) can determine thecontact radii for these two indenters.

To obtain the exact expressions for the 3D coupled field in the half-space, theresults in [41–44] should be further employed. The exact expressions of the couplingphonon–phason fields under indentation are given in Appendix 3 and some keyrelations concerning the indentation responses are given below for the three commonindenters.

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4.1. Flat-ended indenter

Substituting �j ðt, r, �Þ for flat-ended indenter in Equation (16) into Equation (14)

leads to expressions of the phonon field pressure and phason field pressure as

�zz ¼ �p1 rð Þ ¼ ��11dþ �21’0

�2�

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 � r2p , ð19Þ

Hzz ¼ �p2 rð Þ ¼ ��12dþ �22’0

�2�

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 � r2p : ð20Þ

0.0 0.5 1.0 1.5 2.00

1

2

3

4

5

6

7

8

9

10

11

12

P (

N)

d (μm)

R1= -3 GPa

R1= -1.5 GPa

R1= -0.75 GPa

Indentation force in phonon field (a)

(b)

0.0 0.5 1.0 1.5 2.00

25

50

75

100

125

150

175

200

225

250

275

300

R1= -3 GPa

R1= -1.5 GPa

R1= -0.75 GPa

S (μ

N)

d (μm)

Indentation force in phason field

Figure 2. (colour online) Indentation forces versus indentation depth for different R1 for thespherical indenter, ’0¼ 0mm.


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Therefore, the generalised resultant indentation forces read

P ¼2�11a

��dþ

�21�11

’0

� �, ð21Þ

S ¼2�12a

��dþ

�22�12

’0

� �: ð22Þ

In this case, the indentation depth is

d ¼1

�

��P

2a� �21’0

� �: ð23Þ

0.0 0.5 1.0 1.5 2.00

1

2

3

4

5

6

7

8

9

10

11

12

R2= 2.4 GPa

R2= 1.2 GPa

R2= 0.6 GPa

P (

N)

d (μm)

Indentation force in phonon field

0.0 0.5 1.0 1.5 2.00

255075

100125150175200225250275300325350

R2= 2.4 GPa

R2= 1.2 GPa

R2= 0.6 GPa

S( μ

N)

d (μm)


(a)

(b)

Figure 3. (colour online) Indentation forces versus indentation depth for different R2 for thespherical indenter, �0¼ 0 mm.

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4.2. Conical indenter

Substituting �j ðt, r, �Þ for conical indenter in Equation (16) into Equation (14) leads

to the following expressions of the phonon field pressure and phason field pressure

�zz ¼ �p1 rð Þ ¼ ��11 cot

2��cosh�1

a

r

� �, ð24Þ

Hzz ¼ �p2 rð Þ ¼ ��12 cot

2��cosh�1

a

r

� ��12 d� �a cot =2ð Þ þ �22’0

�2�


It can be seen that the singularity in the phonon field stress �zz at r¼ a has been

removed. However, the phason field stress Hzz is still singular at r¼ a .

0.0 0.5 1.0 1.5 2.00

1

2

3

4

5

6

7

8

9

10

11

12

13

R3= 2.4 GPa

R3= 1.2 GPa

R3= 0.6 GPa

P (

N)

d (μm)


0.0 0.5 1.0 1.5 2.00

255075

100125150175200225250275300325350375

R3= 2.4 GPa

R3= 1.2 GPa

R3= 0.6 GPa

S (μ

N)

d (μm)


(a)

(b)

Figure 4. (colour online) Indentation forces versus indentation depth for different R3 for thespherical indenter, �0¼ 0 mm.


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Consequently, the resultant phonon and phason indentation forces are obtained

by integrating Equations (24) and (25), respectively, in the contact region

P ¼�11a

��dþ

�21�11

’0

� �, ð26Þ

S ¼�12a

��dþ 2

�22�12��21�11

� �’0

� �: ð27Þ

From Equation (17), the contact radius can be obtained as

a ¼2

�dþ

�21�11

’0

� �tan , ð28Þ

0.0 0.5 1.0 1.5 2.0

0

1

2

3

4

5

6

7

8

9

10

11

12

P (

N)

d (μm)

ϕ0 = 0 (μm)

ϕ0 = 0.1 (μm)

ϕ0 = 0.2 (μm)

ϕ0 = 0.3 (μm)

(a)

0.0 0.5 1.0 1.5 2.00

25

50

75

100

125

150

175

200

225

250

275

ϕ0 = 0 (μm)

ϕ0 = 0.1 (μm)

ϕ0 = 0.2 (μm)

ϕ0 = 0.3 (μm)

S (μ

N)

d (μm)



(b)

Figure 5. (colour online) Indentation forces versus indentation depth for different phasondisplacement for the spherical indenter.

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and thus the indentation force can also be written as

P ¼2�11�2�

dþ�21�11

’0

� �2

tan , ð29Þ

S ¼2�12�2�

dþ2�22�12��21�11

� �’0

� �dþ

�21�11

’0

� �tan : ð30Þ

0 500 1000 1500 2000

-1.5-1.4-1.3-1.2-1.1-1.0-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.10.0

σzz

(GPa

)

r (μm)

(a) zzσ distribution

0 500 1000 1500 2000

-32-30-28-26-24-22-20-18-16-14-12-10-8-6-4-20

Ηzz

(MPa

)

r (μm)

(b) Hzz distribution

Figure 6. (colour online) Surface normal stress induced by the flat-ended indenter,a ¼ 1000mm.


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0 200 400 600 800 1000 1200 1400 1600 1800 2000

20001900180017001600150014001300120011001000900800700600500400300200100

in GPazzσ

z (μ

m)

r (μm)

-0.1400-0.1300-0.1200-0.1100-0.1000-0.09000-0.08000-0.07000-0.06000-0.05000-0.04000-0.03000-0.02000-0.01000-0.0050000.000

0 200 400 600 800 1000 1200 1400 1600 1800 2000

20001900180017001600150014001300120011001000900800700600500400300200100

in MPazzH

z (μ

m)

r (μm)

-4.000-3.500-3.000-2.500-2.000-1.750-1.500-1.250-1.000-0.7500-0.5000-0.4500-0.3000-0.2000-0.1000-0.050000.000

Stress contour of szz

Stress contour of Hzz

(a)

(b)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

2000

1800

1600

1400

1200

1000

800

600

400

200

0

in MParzσ

z (μ

m)

r (μm)

-90.00-80.00-70.00-60.00-50.00-40.00-30.00-20.00-10.00-5.0000.0005.00010.0020.0030.0040.00

Stress contour of srz(c)

Figure 7. (colour online) Stresses induced by the flat-ended indenter, a¼ 1000 mm.

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As a direct result of Equations (28) and (29), the relation between the contactradius and indentation load can be determined as

a ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2�P

�11 cot

s: ð31Þ

4.3. Spherical indenter

Substituting �j ðt, r, �Þ for spherical indenter in Equation (16) into Equation (14) givesrise to expressions of the phonon field pressure and phason field pressure as

�zz ¼ �p1 rð Þ ¼ �2�11�2�R

ffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 � r2p

, ð32Þ

Hzz ¼ �p2 rð Þ ¼ �2�12�2�R

ffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 � r2p

��12 d� a2=R

� �22’0

�2�


Identical to the conical indenter case, the singularity in the phonon field stress �zzat r¼ a has been removed while the phason field stress Hzz is still singular there.

The resultant phonon and phason indentation forces are obtained by integratingEquations (32) and (33), respectively

P ¼4�11a

3��dþ

�21�11

’0

� �, ð34Þ

S ¼4�11a

3��dþ

3=2�22�12

��122�11

� �’0

� �: ð35Þ

In this case, the contact radius can be derived from Equation (18) as

P ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR dþ

�21�11

’0

� �s, ð36Þ

0 200 400 600 800 1000 1200 1400 1600 1800 2000

2000

1800

1600

1400

1200

1000

800

600

400

200

0

in MPazrH

z (µ

m)

r (µm)

-1.400-1.300-1.200-1.100-1.050-1.000-0.7500-0.5000-0.2500-0.10000.0000.050000.10000.20000.30000.40000.50000.6500

Stress contour of Hzr(d)

Figure 7. Continued.


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and the corresponding indentation forces also take the following forms

P ¼ P14�113��

R1=2 dþ�21�11

� �’0

� �3=2, ð37Þ

S ¼ P24�123��

R1=2 dþ�21�11

� �’0

� �1=2dþ

3=2�22�12

��122�11

� �’0

� �: ð38Þ

The relation between the contact radius and indentation load reads

a ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2��RP

4�11

3

s: ð39Þ

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

-240

-220

-200

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

σ zz

(GPa

)

r (μm)


0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5-5.0

-4.5

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

Ηzz

(GPa

)

r (μm)

(b) zzH distribution

Figure 8. (colour online) Surface normal stress indcued by conical indenter, a ¼ 2:21 mm.

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0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

in GPazzσ

z (μ

m)

r (μm)

-220.0-200.0-150.0-125.0-100.0-75.00-50.00-25.00-15.00-10.00-5.000-3.000-1.000-0.20000.000

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

in GPazzH

z (μ

m)

r (μm)

-4.500-4.000-3.500-3.000-2.500-2.000-1.500-1.000-0.5000-0.2000-0.1000-0.05000-0.02000-0.010000.000

Stress contour of zzσ

Stress contour of zzH

(a)

(b)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

in GParzσ

z (μ

m)

r (μm)

-12.50-12.00-11.00-10.00-9.000-8.000-7.000-6.000-5.000-4.000-3.000-2.000-1.000-0.5000-0.2000-0.10000.000

Stress contour of rzσ(c)

Figure 9. (colour online) Stresses induced by the conical indenter, a¼ 2.21mm.


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Relations (31) and (39) are results new to the literature, which are of greatsignificance for guiding future experimental studies.

5. Numerical results and discussion

In this section, we give some numerical examples according to the analyticalsolutions obtained in this paper. Firstly, the influences of some parameters on theindentation responses are considered. Secondly, the stress distributions in the phasonand phonon fields under the three indenters are studied. Note that as the completematerial constants of 1D hexagonal QCs are rarely found in the literature, somematerial constants are assumed in this section for the numerical study.

The shear modulus of i-Mg–Zn–Y has been reported as �� 46 GPa, the phonon–phason coupling constant is approximately �0:03 m [25], the phason constants for asimple random tiling model are K1¼ 0.3GPa and K2=K1 � 0:6 [24]. Therefore, thefollowing QC material constants are used for the numerical studies:

c11 ¼ 150 c12 ¼ 100, c13 ¼ 90, c33 ¼ 130, c44 ¼ 50,

K2 ¼ 0:18, K1 ¼ 0:3, R1 ¼ �1:5, R2 ¼ 1:2, R3 ¼ 1:2;ð40Þ

both of them are in GPa. Moreover, the radius of the flat-ended cylindrical indenteris assumed to be 1mm, the half apex angle of the conical indenter is taken to be �=3,the radius of the spherical indenter is assumed to be 1mm.

Example 1: This example shows the effect of the coupling constants Riði ¼ 1, 2, 3Þon the indentation forces when the spherical indenter is considered.

Figures 2–4 show the relations between the indentation forces and indentationdepth for various phonon–phason coupling constants. Because the phonon–phasoncoupling constants are much smaller than the phonon elastic constants, theirinfluence on the indentation phonon force is barely perceptible. However, theinfluence of the coupling constants on the indentation phason force is evident.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

in GPazrH

z (µ

m)

r (µm)

-210.0-190.0-170.0-150.0-120.0-100.0-80.00-70.00-60.00-50.00-40.00-35.00-30.00-20.00-10.000.000

Stress contour of Hzr(d)


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Figure 5 shows the relations between the indentation forces and indentationdepth for different phason displacement prescribed over the contact area. Bothindentation forces in the phonon and phason fields increase with phason displace-ment. Apparently, the force in the phason field is more sensitive to phasondisplacement.

Example 2: The distributions of the normal and shear stresses induced by differentindenters are considered. For simplicity, we set d¼ 2 mm and ’0 ¼ 0:3 mm. The stressdistributions under the three indenters will be studied.

0 10 20 30 40 50 60 70 80 90

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

σ zz(G

Pa)

r (μm)


0 10 20 30 40 50 60 70 80 90

-70

-60

-50

-40

-30

-20

-10

0

Ηzz

(MPa

)

r (μm)

(b) zzH distribution

Figure 10. (colour online) Surface normal stress induced by the spherical indenter,a ¼ 44:79mm.


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0 10 20 30 40 50 60 70 80

80

70

60

50

40

30

20

10

0

in GPazzσ

z (μ

m)

r (μm)

-3.000-2.900-2.800-2.600-2.400-2.200-2.000-1.800-1.600-1.400-1.200-0.8000-0.4000-0.2000-0.10000.000

0 10 20 30 40 50 60 70 80

80

70

60

50

40

30

20

10

0

in MPazzH

z (μ

m)

r (μm)

-59.00-58.00-57.00-55.00-50.00-40.00-30.00-20.00-15.00-10.00-6.000-4.000-2.000-1.000-0.50000.000

Stress contour of zzσ

Stress contour of Hzz

(a)

(b)

0 10 20 30 40 50 60 70 80

80

70

60

50

40

30

20

10

0

in MParzσ

z (μ

m)

r (μm)

-700.0-680.0-650.0-600.0-550.0-500.0-400.0-300.0-200.0-100.0-80.00-50.00-25.00-10.00-5.0000.000

Stress contour of rzσ(c)

Figure 11. (colour online) Stresses induced by the spherical indenter, a¼ 44.79mm.

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Figure 6 shows the surface distributions of �zz and Hzz induced by the flat-endedindenter. It is seen that both the normal stresses in the phonon and phason field tendto infinity at r ¼ a, and become zero when r4 a. Figure 7 shows the contours of thegeneralised stress components �zz, Hzz, �rz and Hzr in the r–z plane. Due to theserious singular behavior at r ¼ a on the surface, the contours of �zz and Hzz areplotted from z ¼ 100 mm. The singularities at the flat-ended indenter contact edge areclearly shown in Figure 7a and b.

Figure 8 gives the surface distributions of �zz and Hzz induced by the conicalindenter. Both the normal stresses in the phonon and phason fields tend to infinite atr ¼ 0, due to the obvious geometric discontinuity of the indenter. Meanwhile, onlythe normal stress in the phason field is infinite at r ¼ a, which is coincident with ourtheoretical model. Figure 9 displays the two-dimensional distributions of �zz, Hzz, �rzand Hzr in the r–z plane. Various singularity locations can be clearly identified.

The surface distributions of �zz and Hzz induced by the spherical indenter aregiven in Figure 10. As expected, the normal stresses �zz and Hzz are continuous andfinite at r ¼ 0. The normal phason stress Hzz exhibit singularity behavior at r ¼ a.The stress contours of �zz, Hzz, �rz and Hzr in the r–z plane are shown in Figure 11.It is shown that the peak value of �zz occurs at r ¼ 0.

In these numerical studies, the phason field constants and the phonon–phasoncoupling constants are taken to be smaller than the phonon field constants. Hence,the phason stresses are smaller than the phonon stresses.

6. Conclusions

A general theory on indentation over a 1D QCs half-space by flat-ended, conical andspherical indenters is developed in this paper, motivated by the demand for a deeperunderstanding of the experimental results from indentation techniques. Thetheoretical derivations follow the previous work for transversely isotropic

0 10 20 30 40 50 60 70 80

80

70

60

50

40

30

20

10

0

in MPazrH

z (µ

m)

r (µm)

-12.00-11.80-11.70-11.50-11.00-10.50-10.00-9.000-8.000-7.000-6.000-5.000-3.000-2.000-1.000-0.50000.000

Stress contour of Hzr (d)



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piezoelectric materials, due to the mathematical similarity between the two problems.Explicit relations between indentation forces and indentation depth or contact radiusare given for three common indenters. These can not only serve as benchmarks fornumerical or approximate solutions, but are also valuable in experimental studies. Inaddition, all physical quantities at any location in the QC half-space are given in theAppendices in terms of elementary functions. Such results will be of great help inunderstanding indentation-induced fracture, phase transition and other unusualbehavior [29].

For the contact problem of multi-field coupling materials, more thorough andexact analyses are required to removing all field singularities, e.g. considering thelateral displacement caused by indenter penetration or complete contact conditionswithout any assumptions.

Acknowledgments

This work was supported by the National Natural Science Foundation of China(Nos. 10725210, 11090333 and 11102171), the National Basic Research Program of China(No. 2009CB623200), and the Fundamental Research Funds for Central Universities(No. 2011XZZX002).

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and A.I. Bykov, Dokl. Phys. 45 (2000) p.363.[29] S.N. Dub, Y.V. Milman, D.V. Lotsko and A.N. Balous, J. Mater. Sci. Lett. 20 (2001)

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Appendix 1

The three eigenvalues siði ¼ 1, 2, 3Þ are the solutions of the following eigen equation:

n1s6 � n2s

4 þ n3s2 � n4 ¼ 0, ðA1Þ

where

n1 ¼ c44 R22 � c33K1

,

n2 ¼ c33 �c44K2 þ R3 þ R1ð Þ2

� �� K1 c11c33 þ c244 � c13 þ c44ð Þ

2� �

,

þ R2 2c44R3 þ c11R2 � 2 c13 þ c44ð Þ R3 þ R1ð Þ½ �,

n2 ¼ c44 �c11K1 þ R3 þ R1ð Þ2

� �� K2 c11c33 þ c244 � c13 þ c44ð Þ

2� �

,

þ R3 2c11R2 þ c44R3 � 2 c13 þ c44ð Þ R3 þ R1ð Þ½ �,

n4 ¼ c11 R23 � c44K2

:

ðA2Þ


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Appendix 2

The material constants appearing in Equation (4) are expressed as

�i1 ¼ c13 þ c33si�i1 þ R2si�i2,

�i2 ¼ R1 þ R2si�i1 þ K1si�i2,

�i3 ¼ 2 c11 � c66ð Þ þ c13si�i1 þ R1si�i2½ �,

ðB1Þ

where

�i1 ¼ �i= �isið Þ, �i2 ¼ ��i= �isið Þ, ðB2Þ

and

�i ¼ m1 �m2s2i , i ¼ �c11K2 �m3s

2i � c44K1s

4i , �i ¼ c11R3 �m4s

2i þ c44R2s

4i ,

m1 ¼ �K2 c13 þ c44ð Þ þ R3 R3 þ R1ð Þ, m2 ¼ �K1 c13 þ c44ð Þ þ R2 R3 þ R1ð Þ,

m3 ¼ �c11K1 � c44K2 þ R3 þ R1ð Þ2, m4 ¼ c11R2 þ c44R3 � c13 þ c44ð Þ R3 þ R1ð Þ:

ðB3Þ

Appendix 3

In this section, the exact expressions for all field variables in the half-space are listed case bycase without details.

3.1. Flat-ended indenter

The complete 3D solution of the 1D hexagonal QCs half-space under a flat indenter is

ur ¼ �2�a

r

X3i¼1

Ai 1�a2 � l21i 1=2

a

" #, wk ¼ 2�

X3i¼1

�ikAi sin�1 a

l2i

� �,

�zl ¼ �2�X3i¼1

�ilAi

a2 � l21i 1=2l22i � l21i

,

�zk ¼ �2�X3i¼1

�iksiAi

l1i l22i � a2 1=2l2i l

22i � l21i ,

�2 ¼ �4�c66X3i¼1

Ai

a2 � l21i 1=2l22i � l21i

�2a

r21�

a2 � l21i 1=2

a

" #( ),

ðC1Þ

where k ¼ 1, 2, l ¼ 1, 2, 3, and

l1i ¼12 rþ að Þ

2þz2i

� �1=2� r� að Þ

2þz2i

� �1=2n o,

l2i ¼12 rþ að Þ

2þz2i

� �1=2þ r� að Þ

2þz2i

� �1=2n o,

Ai ¼X2j¼1

Iij �1jdþ �2j’0

�2�, i ¼ 1, 2, 3ð Þ:

ðC2Þ

By notingz! 0 : l1i ! min a, rð Þ, l2i ! max a, rð Þ ðC3Þ

and using Equations (11) and (24), Equation (14) can be readily recovered. The expressions forother physical variables may also be verified similarly.

880 Y.F. Wu et al.

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3.2. Conical indenter

In accordance with Equations (24) and (25), we divide the complete 3D solution into twoparts: the first part corresponding to that induced by mechanical penetration and indicated bysuperscript I, and the second part by the prescribed phason displacement and indicated bysuperscript II. The results of the first part are

uIr ¼ � cot X3i¼1

AIi

rl2i � 2al1ið Þ l22i � r2 1=2

2r2þ

r

2ln l2i þ l22i � r2

1=2h i(

�zi r

2 þ z2i 1=2

2r�

r

2ln zi þ z2i þ r2

1=2h iþa2

2r

),

wIk ¼ cot

X3i¼1

�ikAIi a sin�1

a

l2i

� �þ l2

2i � a2 1=2

� r2 þ z2i 1=2

� zi ln l2i þ l22i � r2 1=2h i

þ zi ln zi þ r2 þ z2i 1=2h io

,

�Izl ¼ � cot X3i¼1

�ilAIi ln l2i þ l22i � r2

1=2h i� ln zi þ z2i þ r2

1=2h in o,

�Izk ¼ cot X3i¼1

�iksiAIi

l22i � a2 1=2

� r2 þ z2i 1=2

r,

�I2 ¼ �2c66 cot X3i¼1

AIi

2a2 � l22i

a2 � l21i 1=2ar2

þzi r

2 þ z2i 1=2

�a2

r2

" #:

ðC4Þ

and the results of the second part are

uIIr ¼ �2�a

r

X3i¼1

AIIi 1�

ða2 � l21iÞ1=2

a

� �, wII

k ¼ 2�X3i¼1

�ikAIIi sin�1

a

l2i

� �,

�IIzl ¼ �2�X3i¼1

�ilAIIi

ða2 � l21iÞ1=2

l22i � l21i, �IIzk ¼ �2�

X3i¼1

�iksiAIIi

l1iðl22i � a2Þ1=2

l2iðl22i � l21iÞ

,

�II2 ¼ �4�c66X3i¼1

AIIi

ða2 � l21iÞ1=2

l22i � l21i�2a

r21�ða2 � l21iÞ

1=2

a

� � �ðC5Þ

where

AIi ¼

X2j¼1

Iij�1j�, AII

i ¼X2j¼1

Iij�1j d� a� cot =2ð Þ þ �2j’0

�2�i ¼ 1, 2, 3ð Þ: ðC6Þ

3.3. Spherical indenter

Similar to the conical indenter case, the complete 3D solution for the spherical indenter isgiven by two parts:

uIr ¼ �2r

�R

X3i¼1

AIi �zi sin

�1 a

l2i

� �þ ða2 � l21iÞ

1=2 1�l21i þ 2a2

3r2

� �þ2a3

3r2

� �,

wIk ¼

1

�R

X3i¼1

�ikAIi ð2a

2 þ 2z2i � r2Þ sin�1a

l2i

� �þ3l21i � 2a2

aðl22i � a2Þ1=2

� �,


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�Izl ¼4

�R

X3i¼1

�ilAIi zi sin

�1 a

l2i

� �� ða2 � l21iÞ

1=2

� �,

�Izk ¼2r

�R

X3i¼1

�iksiAIi � sin�1

a

l2i

� �þaðl22i � a2Þ1=2

l22i

� �,

�I2 ¼ �8c66

3��Rr2

X3i¼1

AIi �2a

3 þ ðl21i þ 2a2Þða2 � l21iÞ1=2

� �,

ðC7Þ

for the first part, and

uIIr ¼ �2�a

r

X3i¼1

AIIi 1�

ða2 � l21iÞ1=2

a

� �, wII

k ¼ 2�X3i¼1

�ikAIIi sin�1

a

l2i

� �,

�IIzl ¼ �2�X3i¼1

�ilAIIi

ða2 � l21iÞ1=2

l22i � l21i, �IIzk ¼ �2�

X3i¼1

�iksiAIIi

l1iðl22i � a2Þ1=2

l2iðl22i � l21iÞ

,

�II2 ¼ �4�c66X3i¼1

AIIi

ða2 � l21iÞ1=2

l22i � l21i�2a

r21�ða2 � l21iÞ

1=2

a

� � �,

ðC8Þ

for the second part, where

AIi ¼

X2j¼1

Iij�1j�, AII

i ¼X2j¼1

Iij�1jðd� a2=RÞ þ �2j’0

�2�ði ¼ 1, 2, 3Þ: ðC9Þ

882 Y.F. Wu et al.

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Indentation on one-dimensional hexagonal quasicrystals: general theory and complete exact solutions

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Transcript of Indentation on one-dimensional hexagonal quasicrystals: general theory and complete exact solutions