Analysis on spherical cavity expansion in gradient elastic media

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Analysis on spherical cavity expansion in gradient elastic media Shuo Wang , Haoxiang Chen , Chengzhi Qi PII: DOI: Reference:

S0093-6413(20)30015-X https://doi.org/10.1016/j.mechrescom.2020.103486 MRC 103486

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Mechanics Research Communications

Received date: Revised date: Accepted date:

26 October 2019 1 February 2020 6 February 2020

Please cite this article as: Shuo Wang , Haoxiang Chen , Chengzhi Qi , Analysis on spherical cavity expansion in gradient elastic media, Mechanics Research Communications (2020), doi: https://doi.org/10.1016/j.mechrescom.2020.103486

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Highlights: 

A simplified strain gradient model for spherical cavity expansion problem is investigated.



Closed-form expressions for stresses, strains and displacement components are given.



The effects of microstructure on the stress and deformation fields are analyzed.

1

Analysis on spherical cavity expansion in gradient elastic media Shuo Wang1, Haoxiang, Chen1,2*, Chengzhi, Qi2 1.

State Key Laboratory of Disaster Prevention and Mitigation of Explosion and Impact, The Army Engineering University of PLA, Nanjing 210007, China.

2.

Beijing High Institution Research Center for Engineering Structure and New Material, Beijing University of Civil Engineering and Architecture ,Beijing, 100044,China.

Corresponding author: Haoxiang, Chen. Tel/Fax: +86 15210919329/ 86-010-6832 2492; E-mail: [email protected]

Abstract In this paper, an analysis for spherical cavity expansion in elastic media is presented. In order to account for the long-range interactions among microstructures undergoing elastic deformation, a simplified strain gradient elasticity constitutive model, including only one extra internal length scale, is invoked. Analytical solutions of stresses, strains and displacement, containing the length scales of cavity and microstructures explicitly, are given. The influence of underlying microstructure on the macroscopic behavior of the material is investigated. The results show that gradient-dependent stresses components are smaller than their classical counterparts, indicating that the strength of material seems to be higher by considering the influence of microstructures. It is noted that the gradient-dependent maximum hoop stress does not occur at the inner surface in contrast with Lamé solution. When the length scale is trivial to cavity size, the gradient-dependent solutions reduce to conventional ones. The improved cavity expansion model can be used to improve the design of spherical pressure vessel and interpret indentation size effect. Keywords: Strain gradient theory, spherical cavity, size effect, microstructure, closed-form analytical solutions

studied in [9] and the references therein. It is worthy noting many experimental observations show that the maximum circumferential stress does not occur at the inner surface of the cavity under internal compression [10]. Microstructures widely exist in various solid materials with the forms of microcracks, microvoids and material grains. Under external loading, the deformation and evolution of microstructures will have a significant effect on the overall macroscopic behavior of solid materials. Due to the lack of internal length scale associated with underlying microstructures, the conventional cavity expansion theory should to be modified. In general, the most common way is to introduce the high-order spatial gradient of internal variables (such as strains or stresses) into the constitutive equations to account for the long-range interactions between microstructures [11-16]. A linear strain gradient elasticity theory was established by Mindlin and Eshel [13] by assuming that the potential energy of an elastic material point is dependent on its strains and strain gradients. Consequently, extensive investigations on the

1. Introduction Spherical cavity expansion theory has been successfully used to a wide range of engineering applications. It has been proved that the Lamé's solution can provide reliable approximations [1-2]. Bishop et al [3] and Hill [4] firstly applied the conventional cavity expansion theory to predict the indentation hardness of metals. Recent experimental evidences for strong size-dependent strengthening effect have been observed in the examination of mechanical behavior of micro/nano-structures [5-6], and the indentation size strengthening effect for metals with micro-indenter were also reported in [7-8]. However, the Lamé's solution is based on the classical elasticity, which assumes that the stress state at one material point is uniquely determined by strain, temperature and stress history at the point without considering the long-range interactions among material points. Therefore, Lamé's solution cannot capture the indentation size effect. It should be pointed out that the conventional Lamé's solutions are also not applicable to the cavity formations in anisotropic elastic solids, such as fiber-reinforced materials, which was 2

ij, j  0

gradient-dependent cavity expansion theory were conducted [17-21]. It should be pointed out that there are five extra material parameters in the strain gradient elasticity theory of Mindlin and Eshel [13], which are difficult to be determined theoretically and experimentally. The expressions for stresses, strains and displacements are also very complex. Therefore, a simplified strain gradient theory, only involving one extra internal length scale, is desirable to be established. In this paper, the spherical cavity expansion problem is investigated by using a simplified strain gradient elasticity theory. The physically inspired boundary conditions associated with gradient terms are employed. The closed-form solutions of stresses, strains and displacement are given. The effects of underlying microstructures on macroscopic behavior of material are analyzed. In Section 2, the constitutive model for the simplified strain gradient theory is reviewed and the boundary-value problem is formulated in Section 3. In Section 4, influences of microstructures on the stress and deformation field are presented. The main conclusions are summarized in Section 5.

The detailed derivations of Eq. (6) and (7) with a variational approach are referred to [22]. It is worth noting that motivated from internal variable theory, the same constitutive equations represented as Eq. (4) were also derived by Aifantis and his co-workers by assuming that the internal variable obeys complete balance law containing both a "source" and "flux" term, rather than conventional "time-evolution laws". The introduction of "flux" term provides the possibility for considering the long-range interaction among of microstructures. The complete derivation processes could be found in references [15, 23-24]. The simplified strain gradient constitutive equations adopted in this section have been successfully used to explain size effect of borehole problem and zonal disintegration of the surrounding rock around a deep circular tunnel [24-25].

2. Simplified Strain gradient elasticity constitutive model

3. Boundary value problem of spherical cavity expansion

The bulk Helmholtz free energy w for an isotropic linear elastic material is assumed to be dependent on the infinitesimal strains and its spatial gradients by Mindlin and Eshel [13]. For simplicity, in this paper the free energy w is assumed to be equal to

As shown in Fig. 1, a spherical cavity problem with radius r0 in an infinite gradient elastic medium subjected to uniform internal pressure p0, is investigated in this section. The spherical coordinate system (r, , ) is adopted. For spherical symmetric problem, the non-zero components of displacement fields is radial displacement ur.

w wij ,v   1 ii jj  ijij cii,k jj,k 2 where

(3)

ij  kkij 2ij  22(kkij )

(1)

ij (ui, j uj,i ) 2is the infinitesimal strain,  and

 are Lame constants in classical elasticity, v=kk is the first strain invariant, c is the material constants called gradient coefficient. Assuming c = 

2

2 ,where

is

internal length of material, then the free energy can be rewritten as

w wij ,v   1ii jj ijij  ii,k jj,k 2 2 2

(2)

By minimizing the total potential energy w in the whole volume, the equilibrium equations (neglecting the body force) and constitutive equations for the simplified strain theory are obtained as

Fig.1 The configuraion for the spherical cavity expaion problem

The expressions of strain components are given by 3

(4)

r  ur ,     ur r

where C1 and C2 are extra integration constants.

(5)

r

,

1

The equilibrium equation in spherical coordinate system (neglecting the body forces) is

r 2r    0 r r

and constitutive equations in explicit component forms are

   r 2  2   r 2  2 2

where

1

1

(8)

where, I3 2 and K3 2 are modified Bessel function of the

(9)

By considering Eqs. (5) to (9), a fourth-order ordinary differential governing equation in terms of radial displacement ur is obtained, which can be divided into two sets of uncoupled second-order equations in the following form.

first and second kind of the index 3/2. The general solution to the governing equation of simplified strain gradient constitutive model can be divided into two parts: first two terms are their classical elasticity counterparts and the rest are the contributions of gradient terms. It should be noted that when c' 0, Eq. (18) will reduce to the solution of classical elasticity. To determine all the unknown integration constants in Eq. (18), four boundary conditions (two classical boundary conditions r |ra p0 , r |r0 and two

(10)

 '  1 . Letting

2   y  1c'(  2  2   22 )ur r r r r  

(11)

extra boundary conditions) are required. In general, extra boundary conditions associated with gradient terms are obtained from variational approach, which can be rewritten in terms of (high order) derivatives of displacements or strains. Therefore, the physical essences of extra boundary can be regarded as the relationships between high order derivatives of the displacements or strains. However, the solutions resulted from variational consistent boundary conditions are complicated, which usually can only be solved numerically. Hence, an alternative simple approach is adopted for the determination of extra boundary conditions with clear physical interpretations and amenable to experimental measurements, the details of which can be found in [23-27]. And the physically inspired extra boundary conditions can be read as

then Eq. (10) can be rewritten as

2 y  2 y  2y  0 r2 r r r2

(12)

The general solution to Eq. (12) is

y  A2  Br r

(13)

where A and B are integration constants, which are needed to be determined by boundary conditions. Substituting Eq. (13) into Eq. (11) yields

2ur 2 ur  2  1 2   A2  Br 2  x x x  x  r where

x  r c' .

(14)

The general solution to Eq. (14) is

solved as follows

ur  A2  Br C1 1  r  C2 1  r  r  c'   c' 

(17)

(18)

2

where c'  ' 2 ,

  r   r   c'  2r c' K3 2  c'     

ur  A2  Br C1  I3 2  r  C2  K3 2  r  r 2r c'  c'  2r c'  c' 

being the Possion's ratio. 2  in spherical coordinate

2 2   (  2  2   22 )1c'(  2  2   22 )ur 0 r r r r  r r r r 

(16)

then, Eq. (15) can be transformed into



2   2  2  r r r

  r   r   c'  2r c' I3 2  c'     

(7)

  E1  1  ,   E 21 , with E being

the Young's modulus and The Laplacian operator reads

respectively the first order modified spherical

Bessel functions of first and second kind. Considering the following relationship between spherical Bessel function and Bessel function

(6)

r  r 2  2r  22 r 2 

2 are

2ur | r0  0 r2 rr

(15) 4

(19)

Then, integration constants are presented as 1  p0r03 1 (1h)h2 , B  0,C  0,C 1  p0 c' eh A 1 2 2E  Th  ETh 

microstructure is negligible and the Lamé solution gives fair approximations. When the length scale of microstructure is comparable to the cavity size, the microstructural effect on macroscopic mechanical behaviors is significant and the differences between gradient and conventional theories are large. As expected, when internal length h=0, the effect of microstructure vanishes and the solutions to gradient theory will reduce to conventional results. It is noted that the effects of microstructure are significant near the cavity and decrease rapidly along the radial direction.

h  c' r0 ,

where, e is natural exponent function,

Th  13h 6. The corresponding expressions for the components of displacement, strains and stresses are expressed as follows.

ur 

(1) p0  r03  h2 r03 1h(h r )e1rh r0      2 2 2E  r0  r Th r  

(20)

1r r 1r r   3 2 3  r  (1) p0 r03  h r03 1h(h r )e h   1 c' e h  0

E

r 

Th r 

0

 

 2Th r

r0

(21) 1r r   3 2 3     (1) p0 r03  h r03 1h(h r )e h  (22) 2E r T r r 0

 

h



0

1r r0  3 2  r p0 r03 1 h 1h(h r )e h  r0 r    Th  

 

Fig.2 The distribution of radial displacement around the spherical

(23)

cavity

1r r  1r r 3  2     p0 r0 3 1 h 1h(h r )e h  p0 c' e h 0

2r   Th 

r0

0

  2Th r

(24) 4. Parametric sensitivity study of microstructural length In this section, the effects of microstructural length on the overall stress and deformation fields are investigated with the following typical mechanical parameters: internal pressure p0=40Mpa Young's modulus E =20GPa, Poisson's ratio = 0.4. For illustration purpose, the distributions of displacement, strains and stresses only within the region of r/r0=1  5 are plotted. For comparison, five dimensionless internal length scales

(a) radial strain

h  c' r0 =0, 0.05, 0.10, 0.15, 0.20 are chosen in the calculation. In Fig. 2-4, the distribution of stress, strain and displacement components and their dependence on the dimensionless cavity size are investigated. When the length scale of microstructure is trivial compared to the geometric scale of spherical cavity, the influence of

(b) tangential strain Fig.3 The distribution of tangential and radial strain around the spherical cavity

5

the size (strengthening) effect for a spherical cavity subjected to uniform internal pressure p0 is investigated. The closed-form analytical solutions to stresses, strains and displacement, containing the scales of microstructure and cavity size explicitly, are presented. The effect of microstructures on the macroscopic behaviors of the material is investigated by using various internal length scales. It is indicated that the effects of microstructures on the distribution of stresses, strains and displacement are significant when the internal length scale is comparable to the cavity size. When the microstructural length is trivial to the radius of cavity, the gradient-dependent solutions reduce to conventional ones. The gradient-dependent tangential stress is no longer monotone decreasing with the radial coordinate and the maximum value of tangential stress does not occur at the inner surface of cavity, which may provide an explanation of the unusual distributions of tangential stresses near the cavity in [10]. The improved cavity expansion model can be used as a reference for the design of spherical pressure vessel and interpretation of indentation size effect.

(a) radial stress

(b) tangential stress Fig.4 The distribution of tangential and radial stress around the spherical cavity

As shown in Fig. 2, the gradient-dependent displacements are slightly larger than classical displacement, which indicates that the material tends to be more flexible with considering the influence of microstructure. The distributions of radial and tangential strains plotted in Fig.3 show that the magnitude of gradient-dependent radial strain r is smaller than conventional one near the cavity; however, the gradient-dependent tangential strain  is larger. From Fig.4, it is noted that the gradient-dependent stresses are smaller than Lamé solution, which indicates that the admissible stress of the material tends to be higher. The distributions of gradient-dependent stresses provide a possibility for predicting size strengthening effect of spherical cavity expansion problem. For this certain boundary value problem in gradient elastic media, the distributions of gradient-dependent tangential stresses are quite different. The tangential stress is no longer monotone decreasing with the radial coordinate and the maximum value of tangential stress does not occur at the inner surface of cavity.

Conflict of Interest We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. Acknowledgments This work was supported by the National Natural Science Foundation of China (NSFC grant No. 51774018), the “973” Key Research Program (grant No. 2015CB0578005), Program for Changjiang Scholars and Innovative Research Team in University (IRT_17R06). References [1] Timoshenko, S., Goodier, J. N., 1970. Theory of Elasticity. 3rd edition, McGraw-Hill, New York. [2] Yu, H. S., 2000. Cavity expansion methods in geomechanics.

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