Energy release rates for a misfitted spherical inclusion under far-field mechanical and uniform thermal loads

European Journal of Mechanics A/Solids 49 (2015) 169e182

Contents lists available at ScienceDirect

European Journal of Mechanics A/Solids journal homepage: www.elsevier.com/locate/ejmsol

Energy release rates for a misfitted spherical inclusion under far-field mechanical and uniform thermal loads S.Y. Seo a, D. Mishra a, b, C.Y. Park a, S.-H. Yoo a, Y.E. Pak b, * a b

Department of Mechanical Engineering, Ajou University, Suwon 443-729, South Korea Advanced Institutes of Convergence Technology, Seoul National University, Suwon 443-270, South Korea

a r t i c l e i n f o

a b s t r a c t

Article history: Received 24 January 2014 Accepted 4 July 2014 Available online 15 July 2014

A misfitted spherical inclusion embedded in an infinite matrix under far-field triaxial mechanical and uniform thermal loads is analyzed in three-dimensional linear elasticity. Closed-form expressions for each loading case have been obtained separately utilizing proper continuity and far-field loading conditions. The principle of superposition has been utilized to obtain the complete solution from these three individual solutions. The solutions are discussed in terms of stress distribution, magnitude and locations of stress concentrations. Closed-form expressions for energy release rates have been derived based on the stress and strain solutions by employing the path-independent integrals, J, L and M considering the inclusion as a defect. The measure of self-similar expansion energy release rate provided by the M-integral brings some interesting insight which can be applicable to many practical cases such as phase transformations in metals and nanoscale defects. The fundamental work presented here will help to better understand the behavior of materials with a misfitted inclusion subjected to various loading environments. © 2014 Elsevier Masson SAS. All rights reserved.

Keywords: Spherically symmetric misfitted inclusion Thermomechanical load Path-independent J-, L- and M-integrals

1. Introduction A situation of misfitted spherical inclusion embedded in a matrix arises in many engineering applications. Some examples are inclusions in composite materials, precipitates in metals and nanostructures in electronic devices. It is important to understand the effects of such precipitates, inclusions or defects in materials as they play an important role in influencing the elastic, piezoelectric and even electronic properties of these materials. Point defects such as vacancies or interstitials in the form of inclusions and cavities have important role in the performance of electronic devices. The strength of materials is the most important criterion for structural applications and machine component design. Therefore, it is important to understand the distribution of stress and strain, and the locations of stress concentrations to predict failure of such systems in presence of various types of defects. There are numerous works available in literature presenting various types of studies and investigations on inclusions embedded in an infinite matrix. Eshelby (1957, 1959) first calculated deformation field associated with an ellipsoidal inclusion in a state of homogeneous strain * Corresponding author. Tel.: þ82 31 888 9056; fax: þ82 31 888 9040. E-mail address: [email protected] (Y.E. Pak). http://dx.doi.org/10.1016/j.euromechsol.2014.07.002 0997-7538/© 2014 Elsevier Masson SAS. All rights reserved.

within an infinite matrix. Many researchers thereafter utilized the concept and generated useful understanding on the problem of inclusions in an infinite matrix. Sankaran and Laird (1976) provided the deformation field for a misfitted spheroidal inclusion. Lee et al. (1999) investigated thermal stresses in the spheroidal inhomogeneity based on the equivalent inclusion approach proposed by Eshelby. The stress concentration problem for transversely isotropic elastic medium containing an arbitrary spheroidal inclusion under tension has been investigated by Kirilyuk and Levchuck (2005). Stress concentration around a spherical cavity in a cubic medium has been examined by Chiang (2007) utilizing the equivalent inclusion approach. The emergence of nanotechnology has brought the subject of inclusions in the forefront of research lately. Some researchers have examined stress concentrations considering the interface stresses around a nanoscale spherical cavity (Li et al., 2006; Lim et al., 2006; Ou et al., 2009). The most recent review on inclusions in an infinite space, half space under surface loading and in a half space under contact loading has been presented by Zhou et al. (2013). These authors have discussed about the literatures on various problems with different types of inclusions under various loading conditions and pointed out the new research directions towards inclusions in a finite space such as spherical inclusion in concentric spherical space.

170

S.Y. Seo et al. / European Journal of Mechanics A/Solids 49 (2015) 169e182

In addition to the stress and strain concentrations, the energy release rate is an important parameter in studying the failure and fracture of engineered materials. There are many studies available in literature on the energy release rates for 2-D inclusions embedded in an infinite matrix. In this regard, most recently, Pak et al. (2012a) provided a closed-form M-integral expression for circular inclusion embedded in an infinite matrix. The effect of material inhomogeneity and their effect on energy release rate has been discussed extensively in that work. In another work, Pak et al. (2012b) also provided closed-form expressions for the pathindependent integrals J, L and M for defects of various shapes embedded in an infinite body under various loading cases. In that work, the authors evaluated the path-independent integrals by first considering the defect of elliptical shape and later using various limiting conditions to obtain the energy release rates for a circular defect and a slit crack. There are numerous works discussing the energetics of nano inclusions in an infinite matrix considering the surface effects and size dependence in two-dimensions (Li and Chen, 2008; Hui and Chen, 2010; Feng and Chen, 2011, 2012). Recently Yu and Li (2013) have proposed new failure theory based on the material configuration forces associated with the M-integral. Elucidation of the energetics of three-dimensional inhomogeneity in an infinite matrix can help in understanding the real material systems with inherent defects such as interstitials, vacancies, inclusions and precipitates. Therefore, we have derived closed-form expressions for path-independent integrals J, L, and M for spherically symmetric misfitted inclusion embedded in an infinite matrix. These path-independent integrals have been evaluated by employing the expressions given by Budiansky and Rice (1973) for three-dimensions. The closed-form expressions provided in this work can be proven handy for material scientists and mechanicians dealing with inhomogeneties and inclusions. The self-similar expansion or contraction of an inclusion provides a very useful physical insight on the energetics and failure mechanism of such systems. Expressions for the elastic fields are simple enough to be found in elasticity texts. However, in this work we present new results, namely, M-integral expression in 3-D for a self-similarly expanding spherical inclusion under various loading cases, which to the best of author's knowledge have not been documented in the literature. 2. Theoretical formulation A misfitted spherical inclusion embedded in an infinite matrix is subjected to a far-field triaxial mechanical load, s∞, and a uniform thermal load, DT, as shown in Fig. 1. The matrix and the inclusion

(oversized by d) considered in this problem are assumed to be linearly elastic and isotropic. This problem gives rise to zero displacement in q and f directions, as it is spherically symmetric. This considerably simplifies the equilibrium equation which can be written as (Bower, 2010)

vsRR 2sRR  sqq  sff þ ¼ 0: vR R

(1)

The strain displacement relations for this case can be expressed as

duR dR uR εqq ¼ R uR ; εff ¼ R εRR ¼

(2)

and the stressestrain constitutive relations become

sRR ¼

 
E EaDT ð1  nÞεRR þ n εqq þ εff  ð1 þ nÞð1  2nÞ ð1  2nÞ

sqq ¼

 
E EaDT ð1  nÞεqq þ n εRR þ εff  ð1 þ nÞð1  2nÞ ð1  2nÞ

sff ¼


E EaDT ; ð1  nÞεff þ nðεRR þ εqq Þ  ð1 þ nÞð1  2nÞ ð1  2nÞ (3)

where, E is Young's modulus, n is Poison's ratio, DT is the uniform thermal load and a is the coefficient of thermal expansion. When the constitutive equation is substituted into the equilibrium equation (1), the governing equation in terms of the radial displacement becomes

 d2 uR 2 duR 2uR d 1 d  2  R ¼ 0: þ  2 ¼ u R R dR dR R2 dR dR R

(4)

When this second order differential equation is integrated twice, the displacement expression becomes

uR ¼ AR þ

B ; R2

(5)

where A and B are constants. The displacement in R-direction is the only variable in the governing equation for the problem at hand. In

Fig. 1. Schematic diagram of misfitted spherical inclusion in an infinite matrix under far-field triaxial and thermal loads.

S.Y. Seo et al. / European Journal of Mechanics A/Solids 49 (2015) 169e182

2 1.5

Spherical Inclusion

σ∞ ΔT δ

Matrix

where superscripts M and I refer to the matrix and the inclusion, respectively. The continuity condition at the interface and the farfield conditions can be written as

σ ∞,ΔT σ ∞, δ ΔT , δ

∞





uIR

¼ uM R

 R¼a  R¼a

uIR  uM ¼ d R

R¼a





sIRR

¼ sM RR

R¼a

R¼a

M sRR

¼ s∞ :

σ ,ΔT, δ

1 Normalized σ

θθ

0.5 0

−0.5

ðwhen there is misfitÞ

(8)

R¼a

−1 0

1

2

3

4

5

R/a Fig. 2. Hoop stress variations along the radial direction for misfitted inclusion embedded in a matrix with various types of loadings when EI/EM ¼ 0.1, aI/aM ¼ 10.

the inclusion, when R becomes zero, the term 1/R2 gives rise to infinite displacement. Therefore, the constant B ¼ 0 has been enforced to eliminate the case of infinite displacement. Hence, the displacement expression for the inclusion becomes

uIR ¼ AR;

(6)

and the expression for the displacement in the matrix can be written as,

uM R ¼ AR þ

171

B ; R2

(7)

Solution for the given problem has been obtained by solving three separate problems, namely spherical inclusion embedded in an infinite matrix subjected to a (i) far-field triaxial mechanical load, s∞, (ii) uniform temperature change, DT, and (iii) misfitted inclusion in an infinite matrix, all in the framework of three dimensional linear elasticity. These three separate problems have been solved based on the governing equation discussed above by properly considering the constitutive equations. Appropriate continuity and far-field conditions for each loading case have been taken into account while solving the problem. The individual solution of each separate problem is provided in Appendix A. The superposition principle has been employed to obtain the final solution adding these three separate solutions (Fig. 1). 2.1. Superposed final solution The summation of displacements, strains and stresses for each problem gives the following results. In the inclusion:

   io  h   R n M 2E 1  2nI aaM DT  d þ a 3 1  2nI 1  nM s∞ þ EI aI DT 1 þ nM D1 a    io  h   1 n M 2E 1  2nI aaM DT  d þ a 3 1  2nI 1  nM s∞ þ EI aI DT 1 þ nM εIRR ¼ εIqq ¼ εIff ¼ D1 a    o io 1 n I hn  E 3 1  nM s∞  2EM aI  aM DT a  2EM d ; sIRR ¼ sIqq ¼ sIff ¼ D1 a uIR ¼

(9)

in the matrix:

 !     i   oa2 1  2nM s∞ 1 þ nM n h M  I I M ∞ I M I M I M a E 1  2n  E 1  2n s a DT þ E ¼ a DT þ þ aE E  a E d R þ D1 EM EM R2 !        i   o a2 2 1 þ nM n h M  1  2nM s∞ M I I M ∞ I M I M I M a E 1  2n  E 1  2n s a DT þ E ¼ a DT þ þ aE E  a E d  εM RR D1 E M EM R3   !     i   oa2 1  2nM s∞ 1 þ nM n h M  M M I I M ∞ I M I M I M a E 1  2n  E 1  2n s a DT þ E ¼ ε ¼ a DT þ þ aE E  a E d þ εM qq ff D1 EM EM R3   i   oa2 2 n h M ∞ a E 1  2nI  EI 1  2nM s∞ þ aEI EM aI  aM DT þ EI EM d sM RR ¼ s  D1 R3 n h    i   oa2 1 M ∞ a EM 1  2nI  EI 1  2nM s∞ þ aEI EM aI  aM DT þ EI EM d ; sM qq ¼ sff ¼ s þ D1 R3 

uM R

M

(10)

172

S.Y. Seo et al. / European Journal of Mechanics A/Solids 49 (2015) 169e182

3

Spherical Inclusion

I

M

I

M

I

M

I

M

I

M

I

I

M

have been normalized with respect to the inclusion thermal stress as considered for the pure thermal loading case (Figs. 2e4). The case, where the oversized inclusion under far-field triaxial load, s∞, has been considered, the stress plot has been normalized with respect to the far-field load. The strain plots also have been normalized similarly to the stress plots where the normalization factors are respective strains in place of stresses described previously.

E /E = 0.1, α /α = 10

Matrix

E /E = 1, α /α = 10 M

E /E = 1, α /α = 0.1

2

I

M

E /E = 10, α /α = 0.1

Normalized σθθ 1 0

3.1. Stress distribution −1 0

1

2

3

4

5

R/a Fig. 3. Hoop stress variations along the radial direction for misfitted inclusion embedded in a matrix under s∞ ¼ 100 MPa, DT ¼ 200  C and misfit d ¼ 0.2% of a for different EI/EM and aI/aM.

where

    D1 ¼ EI 1 þ nM þ 2EM 1  2nI :

3.1.1. Case I: matrix with spherical cavity under far-field triaxial load, s∞ We would first like to discuss the stress distribution in the matrix region when spherically symmetric cavity is embedded in an infinite matrix under various loading conditions. The state of stress in this case can be obtained by substituting EI ¼ nI ¼ 0 in Eq. (10). The closed-form expression for both radial and hoop stresses in the matrix can be given as

s∞ R3  a3 ¼ 1 þ 3 s∞ : 2R 

sM RR ¼ 3. Stress and strain distribution From the perspective of ensuring structural integrity, it is important to know the pattern of stress distribution and the location of stress concentration. We have studied the stress distribution pattern for various combinations of loading and material constants in the inclusion and in the matrix region. We have also studied the stress distribution at the interface of misfitted spherical inclusion and matrix with different EI/EM and aI/aM ratios for various loading conditions. Similar sets of parameters have been chosen to study the strain variations as well. The matrix material properties throughout the analysis have been used as EM ¼ 200 GPa, aM ¼ 2.4  105/ C and nM ¼ 0.3. Poisson's ratio of the inclusion has also been taken to be the same as that of the matrix wherever necessary. A far-field triaxial mechanical load, s∞ ¼ 100 MPa, and a uniform temperature change, DT ¼ 200  C, have been used in the analysis unless stated otherwise. The stress plot for pure far-field mechanical load has been normalized with respect to the far-field load, s∞ (Fig. 2). The stress plot for pure thermal loading case has been normalized with respect to the thermal stress, EIaIDT/(12nI), which is experienced by the freely expanding inclusion when it is embedded in a rigid infinite matrix. In case of oversized inclusion, it has been normalized by EI(d/a)/(12nI). The stress plots for combined loading cases, where uniform thermal load is applied along with other loadings, A,αI/αM=0.1

B,α /α =0.1

A,αI/αM= 1

B,α /α = 1

I

4

M

A,α /α = 10

I

M

I

M

I

M

A

B

B,α /α = 10

2 Normalized σ θθ 0 −2

a3

(11)

Stress in the matrix in this case is only due to the far-field triaxial mechanical load, and the highest stress concentration is 1.5 at the edge of the spherical cavity. 3.1.2. Case II: matrix with spherical inclusion under far-field triaxial load, s∞ (DT ¼ d ¼ 0) The aim of this section is to discuss the nature of stress distribution in the matrix and inclusion when only a far-field triaxial mechanical load is applied. Fig. 2 provides information on the variation of normalized hoop stress in the matrix and the inclusion region in the radial direction. The stresses in the inclusion region have been found to be constant irrespective of the normalized distance R/a. When the inclusion is softer than the matrix (EI/EM ¼ 0.1), the stress concentration becomes higher in the matrix region at the interface. It decreases moving away from the interface and takes the value of the far-field triaxial mechanical load. The opposite trend can be observed, if we consider the inclusion to be stiffer than the matrix. 3.1.3. Case III: spherical inclusion in the matrix under uniform thermal load, DT (s∞ ¼ d ¼ 0) The aim here is to see the variation in stresses due to thermal expansion coefficient mismatch between the inclusion and the matrix. To achieve this, the far-field triaxial mechanical load and the misfit in the inclusion is set to zero. To understand the sole effect of thermal expansion coefficient mismatch between the inclusion and the matrix, we set the elastic properties of the inclusion and matrix to be the same (EI ¼ EM ¼ E, nI ¼ nM ¼ n). When this condition is enforced along with setting the far-field triaxial mechanical load (s∞) and misfit (d) to be zero in Eqs. (9) and (10), the expressions for stresses in both inclusion and matrix can be expressed as

 2E aI 1  M aM DT 3ð1  nÞ a   I 2E a a 3 1  M aM DT sM RR ¼ 3ð1  nÞ R a  a 3 I E a M 1  M aM DT sM : qq ¼ sff ¼ 3ð1  nÞ R a sIRR ¼ sIqq ¼ sIff ¼

−4 −6 0

sM qq

1

2.5

5 I M E /E

7.5

10

Fig. 4. Hoop stress variations at the interface of misfitted inclusion and infinite matrix with various ratios of EI/EM for different aI/aM when s∞ ¼ 100 MPa, DT ¼ 200  C and misfit d ¼ 0.2%.

(12)

S.Y. Seo et al. / European Journal of Mechanics A/Solids 49 (2015) 169e182

It is obvious that the stresses in the inclusion are a function of the thermal expansion coefficients of both the matrix and the inclusion as well as of the uniform temperature change, DT. Fig. 2 presents the normalized hoop stress variation in the radial direction for pure thermal loading case when the inclusion is elastically softer (EI/EM ¼ 0.1) and thermally more expandable (aI/aM ¼ 10). In this case, hoop stress in the inclusion becomes compressive. This is because with higher thermal expansion coefficient of the inclusion, it tries to expand more which is resisted by the matrix, and therefore, it experiences compressive stress exerted by the matrix. The matrix on the other hand resists the expansion of the inclusion, experiences high tensile hoop stress. The opposite trend can be observed when the inclusion with smaller thermal expansion coefficient is embedded in an infinite matrix under uniform thermal load. 3.1.4. Case IV: spherical inclusion misfitted by d embedded in matrix (s∞ ¼ DT ¼ 0) Here, we would like to see the effect of misfit on the stress distribution in the inclusion and in the matrix. Therefore, the elastic properties of the inclusion and the matrix have taken to be the same (i.e., EI ¼ EM ¼ E and nI ¼ nM ¼ n) and the far-field triaxial mechanical load (s∞) and uniform thermal load (DT) are set to be zero in Eqs. (9) and (10). When this condition is enforced, the expressions for stresses in both inclusion and matrix become

 2E d 3ð1  nÞ a    3 d a

sIRR ¼ sIqq ¼ sIff ¼ 

2E 3ð1  nÞ a R    3 E d a ¼ : 3ð1  nÞ a R

sM RR ¼  M sM qq ¼ sff

(13)

We can see from these expressions that when the inclusion becomes oversized with respect to the hole, it experiences compressive hydrostatic stress as expected, while there is tensile hoop stress in the matrix. Fig. 2 provides stress variation in the inclusion and the matrix region as a function of R/a for EI/EM ¼ 0.1 and aI/aM ¼ 10. It is evident that the stress in the inclusion has been found to be compressive as it is softer than the matrix and also oversized which is being compressed by the stiffer matrix. Opposite nature of the stress variation can be observed in the matrix. In the absence of far-field load and the thermal load, it becomes zero far away from the interface. 3.1.5. Case V: spherical inclusion embedded in matrix with s∞, DT and d We now consider different combinations of loadings, i.e., (i) combined mechanical and thermal load, (ii) far-field triaxial mechanical load with misfitted inclusion, (iii) uniform thermal load with misfitted inclusion, and finally (iv) combination of all three loads. The elastic constant of the inclusion is taken to be ten times softer than the matrix while the thermal expansion coefficient of the inclusion is ten times larger than the matrix for all cases (Fig. 2). The stress variation shown in Fig. 2 for this case has been normalized by the thermal stress experienced by the freely expanding inclusion embedded with the rigid matrix as discussed above. Among the different loadings considered for this work, the stress in the inclusion, EIaIDT due to thermal load, DT ¼ 200  C is (2400 MPa), which is much bigger than the stress due to misfit ((d/a)EI ¼ 400 MPa) and the pure far field mechanical load (100 MPa). Therefore, the normalized stresses for combined loading cases in this figure or Figs. 3 and 4 show the stress variations due to far-field mechanical load or the misfit or both in comparison of the thermal load whenever thermal

173

load has been applied along with other loadings. When we consider the mixed thermomechanical loading (DT ¼ 200  C and s∞ ¼ 100 MPa), the inclusion experiences higher compressive stress concentration. This is because under the far-field mechanical load, s∞, with lower elastic constant and higher thermal expansion coefficient, the inclusion tries to deform more which is restricted by the stiffer matrix. In this case, the thermal load becomes predominant, and therefore, net compressive stress concentration becomes higher in the inclusion region which restricts the expansion of the matrix under uniform thermal load, DT. In the matrix region, the magnitude of the hoop stress at the interface becomes tensile which is contributed by both triaxial far-field load, s∞, and thermal load, DT. This stress concentration decreases moving away from the interface. When a far-field triaxial mechanical load, s∞, is applied to the misfitted inclusion in the absence of uniform thermal load, DT, the hoop stress concentration increases at the interface in the matrix side which finally reduces and becomes equal to the far-field mechanical load (Fig. 2). In the inclusion, the stress becomes compressive as the effect of oversized and softer inclusion becomes predominant. A discontinuity in the hoop stress has been observed when there is an oversized inclusion. This is due to the discontinuous displacement field at the interface. Now, if we consider the case of a thermal load with an oversized inclusion, it has been observed that the stress concentration increases in the matrix at the interface due to predominance of the thermal load which again decreases moving far away from the interface. In the inclusion, the compressive stress concentration increases as both misfit and thermal load produces compressive stress. When we consider the combination of all these loads, i.e. when a thermomechanical load is applied to the spherical misfitted inclusion embedded in an infinite matrix, the net stress concentration at the interface becomes higher in the matrix due to the contribution of all these loads. This is true for inclusion as well, in which case the net compressive stress concentration becomes highest. The plot has been normalized by the thermal stress as discussed above, therefore the contribution of far-field mechanical load and the misfit with respect to the inclusion thermal stress can be observed in Fig. 2. We would like to discuss the effect of material properties of the inclusion and the matrix on the stress distribution along the radial direction as well as at the interface when a thermomechanical load is applied to the misfitted inclusion. These plots have been normalized to the inclusion thermal stress as discussed above. As shown in Fig. 3, the highest hoop stress concentration in the inclusion has been observed when we let the thermal expansion coefficient of the inclusion become 10 times smaller than the matrix while keeping the elastic constant of the matrix and inclusion to be the same. Highest compressive stress in the inclusion has been observed when it becomes 10 times softer and 10 times more expandable than the matrix. In the matrix region, highest tensile stress concentration has been observed when it is 10 times stiffer than the inclusion and have 10 times smaller thermal expansion coefficient. The highest compressive stress in the matrix is found when its thermal expansion coefficient is considered 10 times higher than the inclusion. Fig. 4 provides the normalized hoop stress distribution in the matrix at the interface with elastic constant ratio EI/EM for various combination of thermal expansion coefficient ratios under a combined loading case. The highest stress concentration in both regions at the interface has been observed when the inclusion is elastically stiffer with 10 times higher thermal expansion coefficient. In this case, the inclusion experiences tensile stress while matrix goes through compressive stress. Therefore, we can conclude that the thermal expansion coefficient determines the nature (compressive or tensile) while the elastic constant contributes on the magnitude of the stress.

174

S.Y. Seo et al. / European Journal of Mechanics A/Solids 49 (2015) 169e182

3.2. Strain distribution Studying the strain distribution is important for composite materials with inclusions which have numerous applications. Here, the major interest is the transfer of strain from one region to another for different combinations of material properties of the inclusion and the matrix along with different loading conditions. In recent technological development of nanotechnology such as quantum dots or wires, the information on strain fields is very important because they can influence the electronic properties (Maranganti and Sharma, 2007). Fig. 5 provides the normalized hoop strain distribution in the inclusion and in the matrix along the radial direction under various types of loading for the case of EI/EM ¼ 0.1 and aI/aM ¼ 10. When we consider a spherically symmetric inclusion embedded in an infinite matrix under a far-field triaxial mechanical load with elastically softer inclusion, the hydrostatic deformation has been observed to be high in the inclusion. When a pure thermal load is considered for this combination of material properties, the hoop strain has been found to be bigger in the inclusion which decreases moving away from the interface in the matrix region. This is because the thermal expansion coefficient of the inclusion is bigger which facilitates its expansion under uniform thermal load, DT, however the matrix with smaller thermal expansion coefficient restricts the expansion of the inclusion. When we consider a pure misfit in the absence of triaxial mechanical and uniform thermal loads, a higher compressive strain has been observed in the inclusion while higher tensile hoop strain is observed in the matrix with discontinuity at the interface due to the misfit. This is evident as the softer inclusion is being pushed by the stiffer matrix, therefore, it experiences a higher compressive strain. When combined thermomechanical load (s∞, DT) is applied to this system, a higher magnitude of strain in the inclusion is observed for an elastically compliant inclusion because of higher thermal expansion coefficient in the inclusion. Therefore, the strain transfer occurs from the inclusion to the matrix for this case of material combination. When a triaxial mechanical load is being considered with the misfitted inclusion which is elastically softer but has higher thermal expansion coefficient, a higher compressive hoop strain has been observed in the inclusion, while the matrix experiences higher tensile strain. This is because the compliant misfitted inclusion is compressed more due to the stiffer matrix and the far-field triaxial mechanical load. The combination of thermal load and the misfit produces higher strain in the inclusion while matrix experiences very small strain. Therefore, the strain transfer takes place from the inclusion to the matrix for the combination of material properties considered here. Now, when we consider combined thermomechanical load for a misfitted inclusion embedded in an infinite matrix, again the higher strain has been

0.5

Spherical Inclusion

σ∞ ΔT δ

Matrix

σ ∞,ΔT σ ∞, δ ΔT , δ

σ ∞, ΔT, δ

0 Normalized ε θθ

EI/EM = 0.1, αI/αM = 10

EI/EM = 1, αI/αM = 0.1

EI/EM = 1, αI/αM = 10

I

M

I

M

EI/EM = 10, αI/αM= 10

E /E = 10, α /α = 0.1

12 10

Matrix

Spherical Inclusion

8 Normalized ε

θθ

6 4 2 0 0

1

2

3

4

5

R/a Fig. 6. Strain variation in the case of misfitted inclusion in a matrix with R/a for different EI/EM and aI/aM when s∞ ¼ 100 MPa, DT ¼ 200  C and misfit d ¼ 0.2%.

found in the matrix with evident discontinuity due to a misfit. Here, again the thermal expansion coefficient of the inclusion and the matrix along with a uniform thermal load play important role on the magnitude of strain for both regions. Fig. 6 presents the strain distribution in the inclusion and the matrix along the radial direction for various combinations of elastic constant ratios and thermal expansion coefficient ratios for a farfield triaxial load, s∞ ¼ 100 MPa, a uniform thermal load, DT ¼ 200  C and misfit d ¼ 0.2%. When the inclusion is elastically 10 time softer and has 1/10 thermal expansion coefficient than the matrix, highest tensile hoop strain has been observed in the matrix. The inclusion also experiences higher strain in this case. This is because, the matrix with higher thermal expansion coefficient expands more which can not be resisted by the softer inclusion. As the inclusion becomes stiffer, the strain in the matrix at the interface becomes smaller which becomes 10 times far from the interface. In this case, the stiffer inclusion resists the deformation in the matrix. When thermal expansion coefficient of the inclusion becomes 10 times greater than the matrix, lowest strain has been observed in both the matrix and the inclusion, when it becomes 10 times softer than the matrix. This increases as the inclusion becomes stiffer. 4. Energy release rates It is well known that the energy release rates associated with the translation, rotation and self-similar expansion of defects in solids are expressed by the path-independent integrals J, L, and M respectively. The path-independent J-integral, which was introduced by Rice (1968), provides a means of evaluating energy release rates and stress intensity factors. Other path-independent integrals, L and M, were proposed independently by Gunther (1962) and Knowles and Sternberg (1972). All these path-independent integrals are useful in defect mechanics in that they are related to energy release rates. Many researchers utilized these concepts to evaluate stress intensity factors and energy release rates according to the need of the problem. Budiansky and Rice (1973) modified these pathindependent integrals for three-dimensional applications as

Z

  Wnk  Ti ui;k ds

JK ¼

−0.5

EI/EM = 0.1, αI/αM = 0.1

s

Z

−1 0

LK ¼ 1

2

3

4

5

s

Z 

R/a Fig. 5. Strain variation in the case of misfitted inclusion in a matrix in radial direction when EI/EM ¼ 0.1, aI/aM ¼ 10 for various types of loadings.

  ekij Wxj ni þ Ti uj  Tl ul;i xj ds

MK ¼ s

1 Wxi ni  Tj uj;i  Ti ui ds: 2

(14)

S.Y. Seo et al. / European Journal of Mechanics A/Solids 49 (2015) 169e182

Here, s is a closed surface with outer normal ni, W is the strain energy, Ti is the traction, ui is the displacement and xi is the position vector. We have transformed these surface integrals to spherical coordinate to evaluate the energy release rates for different loading cases on the surface as shown in Fig. 7. The path-independent J- and L-integrals were calculated to be zero because the defect, which has a spherical symmetry, can neither translate nor rotate under the given loads. In other words, the total elastic energy is invariant to the translation or the rotation of the inclusion for the problem at hand. The path-independent M-integral, which is associated with the energy release rate for self-similarly expanding defect or a selfsimilar expansion force, has been evaluated using Eq. (15) on a spherical surface as shown in Fig. 7

Z2p Zp  M¼ 0

1 WR  sRR uR;R R  sRR uR R2 sin qdqdf: 2

  2 MB ¼ 2EI EM aM aI  aM h    n   MC ¼ EM 3EI 1  nM aI  aM þ 2aM EM 1  2nI  oi  EI 1  2nM   MD ¼ 3EI EM 1  nM 2

ME ¼ 2EI EM aM :

(15)

0

4.1. M-integral for various loading cases

The M-integral has been evaluated inside the inclusion also to test the correctness of the result which rightly come out to be zero. This is because there is no defect surrounded by the enclosed surface on which M-integral has been evaluated. The closed-form expression of M-integral obtained by evaluating Eq. (15) on the surface in the matrix surrounding the inclusion as shown in Fig. 7 can be given as

" (h 

   6pa2 3 1  nM s∞ þ 2EM aM DT M I

    1  2n a E M¼ M I E E 1 þ nM þ 2EM 1  2nI ) #  i   I M ∞ I M I M I M s þ E E a  a DT þ E E d :  E 1  2n (16) Equation (16) can be simplified and expressed in terms of the interactions between the various loadings as

h   i 2 6pa2 a MA s∞ þ MB DT 2 þ MC s∞ DT þ MD s∞ d þ ME DTd

    M¼ EM EI 1 þ nM þ 2EM 1  2nI (17) where,

175

h    i  MA ¼ 3 1  nM EM 1  2nI  EI 1  2nM

The M-integral provides a measure of self-similar expansion force on defects subjected to various loading conditions and material parameters. We examine the M-integral for various combination of loads, namely, Case I: spherical cavity embedded in an infinite matrix subjected to s∞ and DT, Case II: spherical inclusion under s∞, Case III: spherical inclusion under DT, Case IV: Spherical inclusion misfitted by d embedded in the matrix, Case V: spherical inclusion under s∞ and DT, Case VI: spherical inclusion misfitted by d under s∞, Case VII: spherical inclusion misfitted by d under DT, Case VIII: spherical inclusion misfitted by d under s∞ and DT with varying material constant ratios EI/EM and aI/aM. The matrix material properties throughout the analysis have been used as EM ¼ 200 GPa, aM ¼ 2.4  105/K and nM ¼ 0.3. The Poisson's ratio of the inclusion has been set to be the same as the matrix wherever necessary. A far-field triaxial mechanical load, s∞ ¼ 100 MPa, and a uniform temperature change, DT ¼ 200  C have been used in the analysis unless stated otherwise. To better understand the results of M-integral values presented here for various cases, it will be helpful to first understand the energetics of the considered elastic system. The M-integral is the configurational material force, also called the self-similar expansion force that is equal to the negative of the change in total energy of the system, i.e., the energy of the elastic structure and the loading mechanism (Kienzler and Herrmann, 2000) as shown in Fig. 8. This can be expressed as

 M¼

II I ETotal  ETotal



Da=a

:

(18)

The total energy change of the system can be expressed as

dETotal ¼ dEint þ dEext ;

(19)

where dEint is the internal energy of the system while dEext is the energy due to the work done by the loading mechanism. The work done by the loading mechanism always contributes to the potential energy drop, and therefore, the energy due to dEext is always negative regardless of the loading direction. In a linear elastic system, the work done by the external loading is twice the internal strain energy. Therefore, the M-integral can be expressed in terms of the internal strain energy of the system or the potential energy drop of the loading mechanism as

M¼

Fig. 7. Schematic representation of the path of the J-, L- and M-integrals.

vðdEint Þ 1 vðdEext Þ a¼ a: va 2 va

(20)

As shown in Eq. (18), the M-integral basically provides the change in total elastic energy of the system with respect to the

176

S.Y. Seo et al. / European Journal of Mechanics A/Solids 49 (2015) 169e182

self-similar expansion of the defect. Fig. 8 shows that increase in the compliance of the elastic system increases the drop in the potential energy of the loading mechanism. Therefore, whenever the self-similar expansion of the defect is associated with the increase in the flexibility (stiffness) of the elastic system, the M-integral will give a positive (negative) value. It is worth noting that, given the same defect growth, the change in the total elastic energy is the same irrespective of the loading direction. This explains the positive (negative) definiteness of the M-integral with respect to the mechanical load given an elastic system with a defect. 4.1.1. Case I: spherical cavity embedded in an infinite matrix subjected to s∞ and DT The M-integral expression for a spherical cavity of radius a subjected to a far-field triaxial mechanical load, s∞, and a uniform temperature change, DT, embedded in an infinite matrix can be expressed by setting EI ¼ nI ¼ 0 in Eq. (17), which can be expressed as

MSC ¼

h  i  2 3pa3 3 1  nM s∞ þ 2EM aM DTs∞ EM

:

(21)

The misfit phenomenon does not arise here as there is no inclusion. There is no effect of the thermal load on stresses for an open cavity (Eq. (11)), however, it produces strain, and therefore, has a coupling term with the mechanical load in the M-integral expression (Eq. (21)). The M-integral becomes zero when only uniform thermal load, DT, is applied in the absence of far-field mechanical load, s∞. This is because the system can freely expand, and therefore, does not experience any stress. Hence, there is no change in the total elastic energy due to the change in the cavity size. In the problem of a spherical cavity embedded in an infinite matrix, the triaxial mechanical load, s∞, irrespective of its direction

lowers the total elastic energy with increase in the cavity size, i.e., increase in the compliance of the system. Therefore, for the open cavity case the M-integral becomes positive definite irrespective of the mechanical loading direction (Fig. 9). The magnitude of the Mintegral increases with the far-field loading. When the problem of a spherical cavity embedded in an infinite matrix is subjected to a combined thermomechanical load with both either positive or negative s∞ and DT, the sign of the M-integral depends on the signs of each respective load. For example, given a positive mechanical load, i.e., a tensile load, a positive DT or a positive thermal expansion lowers the potential energy of the loading mechanism due to the expansion of the elastic system. This contributes to increase in the energy release rate, hence the positive M-integral. The opposite happens when DT is negative. 4.1.2. Case II: spherical inclusion in matrix subjected to far-field mechanical load, s∞ (DT ¼ d ¼ 0) When we enforce the condition of no thermal load and no misfit (DT ¼ d ¼ 0) in the Eq. (17), we can get the M-integral expression for pure triaxial mechanical loading, which can be expressed as

M

s∞

      2 18pa3 1  nM EM 1  2nI  EI 1  2nM s∞

    : ¼ EM EI 1 þ nM þ 2EM 1  2nI

(22)

Fig. 9 provides the M-integral variation with respect to the farfield triaxial mechanical load, s∞, for different combinations of inclusion and matrix elastic constants (EI/EM). We can see that for a softer inclusion, M-integral is positive and it increases with the magnitude of the far-field load, s∞, irrespective of its direction. The potential energy drop of the loading mechanism will be bigger when the softer inclusion grows self-similarly making the whole system becomes more compliant. This gives rise to a positive M-integral. When the inclusion becomes stiffer than the matrix and the inclusion grows self-similarly, the overall system becomes stiffer, then the total elastic energy change becomes negative, and therefore, M-

Fig. 8. Concept of the M-integral in terms of the total energy (internal energy of the system and energy due to the loading mechanism) of the system.

S.Y. Seo et al. / European Journal of Mechanics A/Solids 49 (2015) 169e182

1

EI/EM=0 I

M

I

M

E /E =0.1

0.5

E /E =1 I

E /EM=10

M−integral[MN⋅m2] 0

−0.5

−1 −100

−50

0 σ∞ [MPa]

50

100

Fig. 9. M-integral variation with far-field triaxial mechanical load, s∞ for different ratios of EI/EM.

integral becomes negative. Here, it is worth noting that the energetics of an elastic system prefers to have a more flexible structure. Therefore, if a defect is softer (stiffer) than the matrix, the growth of the defect results in a positive (negative) energy release rate. 4.1.3. Case III: spherically symmetric inclusion subjected to uniform temperature change, DT (s∞ ¼ d ¼ 0) When we enforce the condition of no far-field triaxial mechanical load and no misfit (s∞ ¼ d ¼ 0) in the Eq. (17), we can get the M-integral expression for a system under uniform temperature change, DT, which can be expressed as

MDT ¼

  12pa3 EI EM aM aI  aM DT 2     : EI 1 þ nM þ 2EM 1  2nI

(23)

It is interesting to note that M-integral expression shown by Eq. (23) is positive (negative) definite with respect to the thermal load, and the sign depends on the relative magnitude of the thermal expansion coefficient of the matrix and the inclusion.

Ms∞ ;DT

177

in this case at the given DT. When the inclusion becomes stiffer (EI/EM ¼ 10) for the thermal expansion coefficient ratio, aI/ aM ¼ 10, the total energy change required is higher for the defect to grow self-similarly. Therefore, the magnitude of the M-integral becomes higher for given DT. When aI/aM ¼ 0.1, i.e., thermal expansion coefficient of the inclusion is smaller than the matrix, the rate of expansion in the inclusion becomes smaller than in the matrix. This condition gives rise to a negative M-integral. When the inclusion is 10 times elastically stiffer (EI/EM ¼ 10) and aI/aM ¼ 0.1, the magnitude of the negative M-integral becomes higher for a given DT.

4.1.4. Case IV: spherically symmetric inclusion misfitted by d (s∞ ¼ DT ¼ 0) When we enforce the condition of s∞ ¼ DT ¼ 0 in Eq. (17), we get M-integral equal to zero. This implies that, when there is a misfitted inclusion without any external load, there is no change in the total elastic energy with respect to self-similar expansion of the defect. This happens because as there is no external loading, the work done due to external loading mechanism is naturally zero. The total energy of the system is only due to the internal energy, namely, the misfit strain energy. The infinite matrix in this case is in a state of pure shear as the sum of sRR, sqq and sff is zero indicating that there is no net volume change. Therefore, the matrix cannot accommodate self-similarly expanding defect, and the M-integral rightly comes out to be zero. 4.1.5. Case V: spherically symmetric inclusion under s∞ and DT (d ¼ 0) When we enforce the condition of no misfit (d ¼ 0) in the Eq. (17), we can get the expression of M-integral for combined far-field triaxial mechanical load, s∞, and uniform temperature change, DT, for the problem at hand. This can be expressed as

" # (       i h  6pa3 M M I I M ∞2 I M2 M I M    3 1  n 1  2n  E 1  2n a DT 2 ¼ M I þ 2E E a  a E s E E 1 þ nM þ 2EM 1  2nI ) ( )#    n    o M I M I M M M I I M ∞ a a þ 2a E 1  2n  E 1  2n þE 3E 1  n s DT

Here, we have no mechanical load but rather a thermal load so instead of considering the overall effective stiffness of the system, we need to examine the overall effective thermal expansion coefficient. Given a thermal load, the system will have a lower elastic energy if it is free to expand or contract. Therefore, the energetics of the system prefers to have a bigger (smaller) volume or defect size where it has a bigger (smaller) thermal expansion coefficient. Eq. (23) also shows that the magnitude of M-integral depends on the relative stiffness of the elastic domain. Fig. 10 presents the variation in the M-integral with uniform temperature change, DT. When the inclusion and the matrix has the same elastic constants, but thermal expansion coefficient of the inclusion is ten times greater than the matrix, the M-integral has been found to be positive and observed to be increasing with absolute value of the uniform temperature change, DT. When the inclusion becomes elastically softer than the matrix (EI/EM ¼ 0.1), while its thermal expansion coefficient is higher (aI/aM ¼ 10), it expands more with smaller change in the total energy of the system. Therefore, magnitude of the M-integral becomes smaller

(24)

I

M

I

M

E /E =0.1, α /α =10

I

M

I

M

EI/EM= 1, αI/αM=10

I

M

I

M

EI/EM= 10, αI/αM=10

−100

−50

E /E =0.1, α /α =0.1 E /E = 1, α /α =0.1 E /E = 10, α /α =0.1

30

I

M

I

M

20 10 2

M−integral[MN⋅m ] 0 −10 −20 −30 −200

−150

0 ΔT[°C]

50

100

150

200

Fig. 10. M-integral variation with uniform temperature load, DT, when far-field triaxial mechanical load, s∞ ¼ d ¼ 0.

178

S.Y. Seo et al. / European Journal of Mechanics A/Solids 49 (2015) 169e182

The influence of each load and the interaction term on the energy release was already discussed previously. Here, we will elaborate on the relative contribution of each term on the M-integral. Fig. 11 provides the M-integral variation in the case of combined thermomechanical load for different combinations of material constants. In this case, when the inclusion has higher thermal expansion coefficient (aI/aM ¼ 10), with increasing elastic constant of the inclusion, M-integral becomes positive except for the short range of DT, when the far-field triaxial mechanical load, s∞ ¼ 100 MPa becomes predominant. This is because when the inclusion becomes stiffer, the total energy of the overall system increases due to dominant triaxial mechanical load, which gives rise to negative M-integral in this range. As DT increases, i.e. the uniform thermal load becomes predominant again, M-integral becomes positive. The obvious reason is, when DT becomes predominant and the thermal expansion coefficient of the inclusion is higher than the matrix, the defect (here, the inclusion) expands more which is favorable for the overall system to move towards lower energy state. Therefore, the M-integral becomes positive. The opposite trend can be seen when the thermal expansion coefficient of the inclusion is smaller (aI/aM ¼ 0.1) for the similar elastic constant of the inclusion (EI/EM from 0.1 to 10) as explained above. This is because when the inclusion is less expandable than the matrix, the inclusion becomes smaller with respect to the infinite matrix under uniform temperature change, DT. This is not a favorable condition, as it increases the total energy of the overall system, and therefore, M-integral becomes negative. When s∞ becomes predominant in the smaller range of DT, the M-integral becomes positive when the inclusion is softer and becomes negative when the inclusion is elastically stiffer. This is because under a dominant far-field triaxial mechanical load, the defect becoming bigger gives total potential energy drop when the inclusion is softer but the total potential energy increases when the inclusion is elastically stiffer. 4.1.6. Case VI: spherical inclusion misfitted by d under s∞ (DT ¼ 0) When we enforce the condition of no thermal load (DT ¼ 0) in Eq. (17), we can get the expression of M-integral for the misfitted inclusion embedded in an infinite matrix under far-field triaxial mechanical load as

Ms∞ ;d

It would be interesting to elaborate on the influence of the interaction between the misfit and the far-field mechanical load on the M-integral. In the presence of a tensile triaxial load, the misfit gets relaxed lowering the internal strain energy, hence a positive M-integral. On the other hand, a compressive load will increase the strain energy resulting in negative M-integral. The influence of the misfit on the drop in the external loading mechanism would be nonexistent as the displacement due to the misfit dies out as 1/R2 in the matrix. A similar behavior is expected for the thermal load as will be shown by Eq. (25). Fig. 12 presents the variation in the M-integral with far-field mechanical load for various cases of misfit when the inclusion is elastically ten times softer than the matrix (EI/EM ¼ 0.1). When there is no misfit, as explained above, the M-integral increases or decreases with the magnitude of the far-field load, s∞. When there is misfit between inclusion and the matrix, M-integral becomes negative when the far-field load is compressive. This is because when the compressive far-field load is applied, the total strain energy of the system increases. When the far-field load becomes positive, the misfit get relaxed as explained above and the M-integral becomes positive. In this case, misfit and the tensile far-field triaxial load both helps the defect to become bigger. The magnitude of the M-integral increases with increasing percentage of the misfit.

4.1.7. Case VII: spherical inclusion misfitted by d in matrix under DT (s∞ ¼ 0) When we enforce the condition of no far-field triaxial mechanical load (s∞ ¼ 0) in Eq. (17), we can get the M-integral expression for the misfitted inclusion embedded in an infinite matrix under uniform thermal load, DT, as

MDT;d ¼



  12pa2 EI EM aM a aI  aM DT 2 þ dDT

    : EI 1 þ nM þ 2EM 1  2nI

(26)

Fig. 13 shows the variation in the M-integral with uniform thermal load for various cases of misfit when elastic and thermal expansion coefficients (EI/EM ¼ aI/aM ¼ 0.1) of the inclusion is ten

i  
    ∞2 18pa2 1  nM a EM 1  2nI  EI 1  2nM s þ EI EM ds∞

    : ¼ EM EI 1 þ nM þ 2EM 1  2nI

EI/EM=0.1, αI/αM=0.1

I

M

I

I

M

I

(25)

M

E /E =0.1, α /α =10

I

M

I

M

E /E = 1, α /αM=10

I

M

I

M

EI/EM= 10, αI/αM=10

E /E = 1, α /α =0.1 E /E = 10, α /α =0.1

30

2

20

1.5

10

1

M−integral[MN⋅m2] 0

2 M−integral[MN⋅m ] 0.5

−10

0

−20

−0.5

−30 −200

−150

−100

−50

0 ΔT[°C]

50

100

150

200

Fig. 11. M-integral variation with uniform temperature load, DT, for different EI/EM and aI/aM when far-field triaxial mechanical load, s∞ ¼ 100 MPa.

−1 −100

δ=0 δ = 0.2% of a δ = 0.3% of a

−50

0 ∞ σ [MPa]

50

100

Fig. 12. M-integral variation with far-field triaxial mechanical load, s∞ for, different cases of misfit when EI/EM ¼ 0.1.

S.Y. Seo et al. / European Journal of Mechanics A/Solids 49 (2015) 169e182

50

179

I

M

I

M

E /E =0.1, α /α = 10

I

M

I

M

E /E = 1, α /α = 10

I

M

I

M

E /E =0.1, α /α =0.1

δ=0 δ = 0.2% of a δ = 0.3% of a

E /E = 1, α /α =0.1 E /E = 10, α /α =0.1

I

M

I

M

I

M

I

M

I

M

I

M

E /E = 10, α /α = 10

30

0

20

M−integral[MN⋅m2]

10

−50

M−integral[MN⋅m2] 0 −10

−100 −200

−100

0 ΔT[°C]

100

200

Fig. 13. M-integral variation with uniform temperature load, DT, for different cases of misfit when EI/EM ¼ 1 and aI/aM ¼ 0.1.

−20 −30 −200

−100

0 ΔT[°C]

100

200

Fig. 15. M-integral variation with uniform temperature load, DT, for different EI/EM and aI/aM when far-field triaxial mechanical load, s∞ ¼ 100 MPa and misfit, d ¼ 0.2% of a.

times smaller than that of the matrix. In this case of loading, the defect (softer inclusion) becomes smaller with respect to the matrix because it is thermally less expandable which makes it smaller with respect to the infinite matrix, hence gives rise to a negative M-integral. The M-integral becomes positive in the small range of positive DT, because the softer inclusion with misfit is favorable for the overall system to become compliant. This trend is reversed when dT again becomes predominant because the inclusion with smaller thermal expansion coefficient is unfavorable for the defect growth. Therefore, M-integral again becomes negative. Now, our aim is to understand the effect of material properties on the M-integral for misfitted inclusion under uniform thermal load, DT, in the absence of far-field triaxial mechanical load, s∞. Fig. 14 shows the effect of material constants on the M-integral variation with uniform thermal load, DT, for the misfitted inclusion. Here, the misfit is equal to 0.2% of the inclusion radius, a. The effect of misfit on the M-integral is more evident when inclusion becomes ten times elastically stiffer than the matrix. When the inclusion is 10 times less expandable than the matrix, the magnitude of the M-integral increases, but it comes out to be negative when DT is predominant. This is because the inclusion is less expandable than the matrix, and therefore, the inclusion becomes smaller with respect to the matrix due to the uniform temperature change as explained above. It becomes positive for the short range of positive DT due to a strong effect of the misfit which contributes to a decrease in the overall energy of the system. This again decreases when DT becomes predominant. The

I

M

I

M

I

M

I

M

E /E =0.1, α /α =0.1 E /E = 10, α /α =0.1

I

M

I

M

I

M

I

M

E /E =0.1, α /α =10 E /E = 10, α /α =10

30 20 10 2

M−integral[MN⋅m ] 0 −10 −20 −30 −200

−100

0 ΔT[°C]

100

200

Fig. 14. M-integral variation with uniform temperature load, DT, for different EI/EM and aI/aM when misfit, d ¼ 0.2% of a.

opposite trend happens when the thermal expansion coefficient of the inclusion is higher than the matrix while it becomes elastically stiffer. The figure shows very small region of negative M-integral for the case when the elastic constant and the thermal expansion coefficient of the inclusion is ten times larger than the matrix. This is because, in this region, effect of misfit supersedes the effect of the uniform temperature change, DT. The misfitted stiffer inclusion increases the strain energy when it becomes bigger, which is not a favorable condition.

4.1.8. Case VIII: spherical inclusion misfitted by d embedded in matrix under s∞ and DT A closed-form M-integral expression for this case has been expressed by Eq. (17). After considering different combinations of loading, our aim now is to understand the effect of s∞, DT and d on the M-integral. Various material constant ratios of the inclusion to that of the matrix have been considered to understand the variation in the M-integral for this case. Fig. 15 presents the variations in the M-integral with the uniform temperature change, DT, when a far-field triaxial mechanical load, s∞ ¼ 100 MPa and a misfit between the inclusion and the matrix is 0.2% of the inclusion radius a. When the inclusion has ten times smaller thermal expansion coefficient than the matrix, i.e. aI/aM ¼ 0.1, with increasing elastic constant of the inclusion the magnitude of negative M-integral increases for given range of DT and s∞ ¼ 100 MPa. The inclusion becomes elastically stiffer which is less likely to expand thermally than the matrix, as a result the defect becomes smaller giving rise to the negative Mintegral. When the thermal expansion coefficient of the inclusion is ten times higher than the matrix, i.e., aI/aM ¼ 10, the inclusion has tendency to expand more under the uniform thermal load. Therefore, self-similar expansion of the defect takes place, giving rise to the positive M-integral. The figure shows very small range of negative M-integral when the range of uniform thermal load is very small for elastically stiffer material. In this range, the misfit between the inclusion and the matrix along with the far-field triaxial mechanical load becomes predominant. Defect becoming bigger under the triaxial mechanical load for misfitted stiffer inclusion is unfavorable for the overall system which always prefer to become more compliant. The misfit increases or decreases the magnitude of the M-integral depending upon the sign of the uniform thermal load, DT. When DT is positive, it contributes in increasing the defect size, i.e. positive M-integral and vice versa for the opposite case as explained above.

180

S.Y. Seo et al. / European Journal of Mechanics A/Solids 49 (2015) 169e182

5. Conclusion This paper provides the closed-form expressions of the displacement, strains and stresses for misfitted inclusion under a far-field triaxial mechanical load, s∞, and a uniform thermal load, DT, for a spherically symmetric inclusion embedded in an infinite matrix. The solution has been found by solving three individual problems in the domain of three dimensional linear elasticity and superimposing them to obtain the combined closed-form expressions for these loading parameters. The stress and strain variations in different loading environments have been discussed in terms of the various ratios of material constants for the inclusion and the matrix. This can aid in understanding the stress and strain distributions in these types of embedded inhomogeneities in the material system, and thus, help to predict the failure of such systems. The closed-form expressions for the path-independent integrals, J, L and M have been obtained by taking a contour in the matrix region considering the spherical inclusion as a defect. The path independent J- and L-integrals have rightly been found to be zero as neither translation nor rotation of the defect takes place. However, the path independent M-integral provides very interesting insights especially when a misfit and a thermal load along with a far-field mechanical load have been applied. It was shown that the energetics of an elastic system prefers to have a more flexible structure. When a pure far-field triaxial mechanical load is applied, the Mintegral becomes positive (negative) when softer (stiffer) inclusion grows self-similarly. In case of the thermal load, the system can move to lower energy state when it is allowed to freely expand or contract. This happens when the thermal expansion coefficient of the inclusion is greater than the matrix. Therefore, the M-integral becomes positive (negative) when the thermal expansion coefficient of the inclusion becomes bigger (smaller) for the considered system under uniform thermal load. The M-integral comes out to be zero when misfitted inclusion embedded in a matrix without any external loading is considered. This is because, the system in this case is in pure shear, and therefore, self-similar expansion of the defect does not take place. When the combined thermomechanical load is applied, it has been found that the thermal expansion coefficient of the inclusion determines the direction of the M-integral while elastic constant of the inclusion decides the magnitude. In the presence of a misfitted inclusion, the sign of the interaction term contributing to the M-integral is determined by the direction of the mechanical and the thermal loads. This is because these loads, depending on the direction, either relieve or enhance the elastic strain energy induced by the misfit. In addition it was found out that the misfit increases (decreases) the magnitude of the M-integral as the inclusion becomes elastically softer (stiffer) and thermally more (less) expandable when either triaxial far-field load or the uniform thermal load or both are applied. Insight obtained through this work can be helpful in understanding phase transformations in metals as well as in engineering the strength of the composite materials with various types of inclusions. This work can also be helpful in designing various nanotechnological applications where these types of inclusions and loadings are present.

Acknowledgment This work was supported by the grant 2013-P3-29 from the Advanced Institutes of Convergence Technology at Seoul National University. The authors would also like to acknowledge the support provided by National Research Foundation, Korea for supporting this work under the grant-2013R1A1A2062216. The authors SYS, CYP and SHY would like to acknowledge the support provided by Ajou University under its internal grant S-2012-G0001-00085.

Appendix A (a) Spherically symmetric inclusion embedded in an infinite matrix under far-field triaxial mechanical load The solution of spherical inclusion embedded in an infinite matrix under a far-field triaxial mechanical load, s∞, can be obtained by letting the thermal load, DT ¼ 0 in the constitutive equation. The expression for displacement, strain and stress components in the inclusion becomes

uIR ¼ A1 R εIRR ¼ εIqq ¼ εIff ¼ A1

(A.1) EI

sIRR ¼ sIqq ¼ sIff ¼   A1 : 1  2nI These expressions in the matrix can be written as

uM R ¼ B1 R þ εM RR ¼ B1 

C1

R2 2C1 R3

M εM qq ¼ εff ¼ B1 þ

sM RR ¼ 

E

M

C1 (A.2)

R3 M

1  2nM

2E  B1    C1 1 þ nM R3

EM EM M B sM þ    C1 1 qq ¼ sff ¼  1  2nM 1 þ nM R3 Three simultaneous equations can be obtained in accordance with the continuity conditions at the interface, and the far-field condition as

A1 a ¼ B1 a þ  

EI 1  2n

I

C1 a2

 A1 ¼ 

EM 1  2nM

EM

2EM  B1    C1 1  2n 1 þ nM a3 M

(A.3)

 B1 ¼ s∞ :

Solving these three simultaneous equation (A.3) gives

   3 1  nM 1  2nI s∞   i; A1 ¼  EI 1 þ nM þ 2EM 1  2nI   s∞ 1  2nM B1 ¼ EM   i    1 þ nM EM 1  2nI  EI 1  2nM s∞ h    i a3 : C1 ¼ EM EI 1 þ nM þ 2EM 1  2nI

(A.4)

The closed form expressions for displacements, strains and stresses in the inclusion and the matrix region can be obtained by substituting A1, B1 and C1 into eqs. (3) and (4) as follows;in the inclusion

S.Y. Seo et al. / European Journal of Mechanics A/Solids 49 (2015) 169e182

   3 1  nM 1  2nI s∞   i R ¼  EI 1 þ nM þ 2EM 1  2nI    3 1  nM 1  2nI s∞   i εIRR ¼ εIqq ¼ εIff ¼  EI 1 þ nM þ 2EM 1  2nI   3EI 1  nM s∞ I I I   i ; sRR ¼ sqq ¼ sff ¼  EI 1 þ nM þ 2EM 1  2nI

A2 a ¼ B2 a þ

uIR

 (A.5)

EI aI DT EM 2EM EM aM DT A ¼   C       B    2 2 2 1  2nI 1  2nM 1  2nI 1  2nM 1 þ nM a3 E

EM

EM aM DT  B2   ¼0 1  2nM 1  2n M

From the above three simultaneous Eq. (A.7), we can evaluate A2, B2 and C2 as

  i    1þnM EM 12nI EI 12nM s∞ h    i Rþ EM EI 1þnM þ2EM 12nI

 3 a  2 R     i    s∞ 12nM 2 1þnM EM 12nI EI 12nM s∞ h    i  εM RR ¼ EM EM EI 1þnM þ2EM 12nI  3 a  3 R      i   s∞ 12nM 1þnM EM 12nI EI 12nM s∞ M M h    i þ εqq ¼ εff ¼ EM EM EI 1þnM þ2EM 12nI  3 a  3 R   i

 2 EM 12nI EI 12nM s∞ a3 ∞   i sM RR ¼ s  I  R3 E 1þnM þ2EM 12nI i s∞ a3 E 12n E 12n M ∞   i sM qq ¼ sff ¼ s þ I  R3 E 1þnM þ2EM 12nI

a2

I

   i EI aI 1 þ nM þ 2EM aM 1  2nI DT   i A2 ¼

I E 1 þ nM þ 2EM 1  2nI

  s∞ 12nM EM

C2

(A.7)

and in the matrix

uM R ¼



181

M



I



I



B2 ¼ aM DT    EI 1 þ nM aI  aM DT   ia3 : C2 ¼  EI 1 þ nM þ 2EM 1  2nI

(A.8)

The closed form expressions for displacements, strains and stresses in the inclusion and the matrix region can be obtained by substituting A2, B2 and C2 into Eqs. (3) and (4);in the inclusion,

   i EI aI 1 þ nM þ 2EM aM 1  2nI DT   i R uIR ¼

I E 1 þ nM þ 2EM 1  2nI   i

I I E a 1 þ nM þ 2EM aM 1  2nI DT I I I   i εRR ¼ εqq ¼ εff ¼

I E 1 þ nM þ 2EM 1  2nI   2EI EM aM  aI DT   i ; sIRR ¼ sIqq ¼ sIff ¼  EI 1 þ nM þ 2EM 1  2nI

(A.9)

and in the matrix,

M

(A.6)

(b) Spherically symmetric inclusion embedded in infinite matrix under uniform thermal load, DT The solution for the case of a thermal loading in terms of uniform temperature change has been obtained by letting the far-field mechanical load, s∞ ¼ 0 in the constitutive equations. Same governing equations (Eqs. (6) and (7)) hold true in this case as well. This can provide the change in displacement, strain and stress variations in the matrix and the inclusion region due to a uniform temperature change. The displacement and stress continuity conditions at the interface of the inclusion and matrix are the same as in the previous case. The continuity conditions at the interface can be expressed as

     EI 1 þ nM aI  aM DT a3 M   i uM R ¼ a DT R þ I  R2 E 1 þ nM þ 2EM 1  2nI      2EI EM 1 þ nM aI  aM DT a3 M   i εM RR ¼ a DT  I  R3 E 1 þ nM þ 2EM 1  2nI      EI 1 þ nM aI  aM DT a3 M M M   i εqq ¼ εff ¼ a DT þ  R3 EI 1 þ nM þ 2EM 1  2nI    3 2EI EM aI  aM DT a    i sM RR ¼  I M M I R3 E 1þn þ 2E 1  2n    3 EI aI  aM DT a M M   i EM : sqq ¼ sff ¼  R3 EI 1 þ nM þ 2EM 1  2nI (A.10)

182

S.Y. Seo et al. / European Journal of Mechanics A/Solids 49 (2015) 169e182

(c) Misfitted inclusion embedded in an infinite matrix

Appendix B. Supplementary data

In this case, the inclusion is bigger than the spherical hole of radius a by d. Either the bigger inclusion needed to be compressed or the smaller spherical hole in the matrix needed to be expanded so that the oversized inclusion can fit. The displacement continuity condition at the interface for this case will be different than previous cases (refer, Eq. (8)), which can be expressed as

A3 a þ d ¼ B3 a þ

C3 a2

I

E EM 2EM A B ¼        C3 3 3 1  2nI 1  2nM 1 þ nM a 3

(A.11)

EM   B3 ¼ 0: 1  2nM Solving three simultaneous equations (Eq. (A.10)) provides the constants A3, B3 and C3 as

  2EM 1  2nI d   i A3 ¼  a EI 1 þ nM þ 2EM 1  2nI B3 ¼ 0 2 I



M



(A.12)

a E 1þn   i : C3 ¼  EI 1 þ nM þ 2EM 1  2nI This provides the effect of misfit on stresses and strains. Eq. (A.12) is substituted into Eqs. (3) and (4) to obtain the expressions for displacements, strains and stresses in the inclusion as

  2EM 1  2nI d   i R uIR ¼   a EI 1 þ nM þ 2EM 1  2nI   2EM 1  2nI d I I I   i εRR ¼ εqq ¼ εff ¼   a EI 1 þ nM þ 2EM 1  2nI

(A.13)

2EM d   i ; sIRR ¼ sIqq ¼ sIff ¼   a EI 1 þ nM þ 2EM 1  2nI and in the matrix

   2 EI 1 þ nM d a   i ¼  R2 EI 1 þ nM þ 2EM 1  2nI    2 2EI 1 þ nM d a    i εM RR ¼ I M M I R3 þ 2E 1  2n E 1þn uM R

εM qq

¼

εM ff

   2 EI 1 þ nM d a   i ¼  R3 EI 1 þ nM þ 2EM 1  2nI

 2 2EI EM d a   i sM RR ¼  I  M M I R3 þ 2E 1  2n E 1þn  2 E I EM d a M    i : ¼ s ¼ sM

I qq ff R3 E 1 þ nM þ 2EM 1  2nI

(A.14)

Supplementary data related to this article can be found at http:// dx.doi.org/10.1016/j.euromechsol.2014.07.002.

References Bower, A.F., 2010. Applied Mechanics of Solids. CRC Press Taylor and Francis Group, Boca Raton, London, New York. Budiansky, B., Rice, J.R., 1973. Conservation laws and energy-release rates. J. Appl. Mech. 40, 201e203. Chiang, C.R., 2007. Stress concentration around a spherical cavity in a cubic medium. J. Strain Anal. Eng. Des. 42, 155e162. Eshelby, J.D., 1957. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. A241, 376. Eshelby, J.D., 1959. The elastic field outside an ellipsoidal inclusion. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 252 (1271), 561e569. Feng, H.Y., Chen, Y.H., 2011. The area contraction and expansion for a nanovoid under four different kinds of loading. Arch. Appl. Mech. 81, 1323e1331. Feng, H.Y., Chen, Y.H., 2012. Energy release or absorption due to simultaneous expansion of many interacting nanoholes in elastic materials. Arch. Appl. Mech. 82, 163e175. Gunther, W., 1962. Uber einige randintegrale der elastomechanik. In: Adhandlungen der Braunschweigischen Wisssenschaftlichen. Gesellschaft, vol. XIV. Verlag Friedr. Vieweg, Braunschweig. Hui, T., Chen, Y.H., 2010. The M-integral analysis for a nano-inclusion in plane elastic materials under uniaxial or biaxial loadings. J. Appl. Mech. 77, 021019-1-9. Kienzler, R., Herrmann, G., 2000. Mechanics in Material Space with Application to Defect and Fracture Mechanics. Springer-Verlag, Berlin. Kirilyuk, V.S., Levchuck, O.Y., 2005. Stress state of a transversely isotropic medium with an arbitrary oriented spheroidal inclusion. Int. Appl. Mech. 41 (3), 137e143. Knowles, J.K., Sternberg, E., 1972. On a class of conservation laws in linearized and finite elastostatics. Arch. Ration. Mech. Anal. 44, 187e211. Lee, S., Liaw, P.K., Liu, C.T., Chou, Y.T., 1999. Thermal stresses due to spheroidal inclusions. Mat. Chem. Phys. 61, 207e213. Li, Q., Chen, Y.H., 2008. Surface effect and size dependence on the energy release due to nano sized hole expansion in plane elastic material. J. Appl. Mech. 75, 061008 (1e5). Li, Z.R., Lim, C.W., He, L.H., 2006. Stress concentration around a nano-scale spherical cavity in elastic media: effect of surface stresses. Eur. J. Mech. A/Sol. 25, 260e270. Lim, C.W., Li, Z.R., He, L.H., 2006. Size dependent non-uniform elastic field inside a nano-scale spherical inclusion due to interface stress. Int. J. Sol. Struct. 43, 5055e5065. Maranganti, R., Sharma, P., 2007. Strain field calculations in embedded quantum dots and wires. J. Comput. Theor. Nanosci. 4, 715e738. Ou, Z.Y., Wang, G.F., Wang, T.J., 2009. Elastic fields around a nano sized spherical cavity under arbitrary uniform loadings. Eur. J. Mech. A/Sol. 28, 110e120. Pak, Y.E., Mishra, D., Yoo, S.H., 2012a. Closed-form solution for a coated circular inclusion under uniaxial tension. Acta Mech. 223 (5), 937e951. Pak, Y.E., Mishra, D., Yoo, S.H., 2012b. Energy release rates for various defects under deferent loading conditions. J. Mech. Sci. Technol. 26 (11), 3549e3554. Rice, J.R., 1968. A path independent integral and the approximate analysis of strain concentration by notches and cracks. J. Appl. Mech. 35, 379e386. Sankaran, R., Laird, C., 1976. Deformation field of a misfitting inclusion. J. Mech. Phys. Sol. 24, 251e262. Yu, N.Y., Li, Q., 2013. Failure theory via the concept of material configurational forces associated with the M-integral. Int. J. Sol. Struct. 50, 4320e4332. Zhou, K., Hoh, H.J., Wang, X., Keer, L.M., Pang, J.H.L., Song, B., Wang, Q.J., 2013. A review of recent works on inclusions. Mech. Mat. 60, 144e158.