Symmetrization of the effective stress tensor for anisotropic damaged continua

Accepted Manuscript

Symmetrization of the effective stress tensor for anisotropic damaged continua Artеm S. Semenov PII: DOI: Reference:

S2405-7223(17)30089-0 10.1016/j.spjpm.2017.09.005 SPJPM 149

To appear in:

St. Petersburg Polytechnical University Journal: Physics and Mathematics

Received date: Accepted date:

12 September 2017 12 September 2017

Please cite this article as: Artеm S. Semenov , Symmetrization of the effective stress tensor for anisotropic damaged continua, St. Petersburg Polytechnical University Journal: Physics and Mathematics (2017), doi: 10.1016/j.spjpm.2017.09.005

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Symmetrization of the effective stress tensor for anisotropic damaged continua Artem S. Semenov Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russian Federation

AN US

CR IP T

A uniform formulation of the effective stress tensor symmetrization procedure has been proposed. This form contains the classical additive and multiplicative symmetrization schemes as particular cases. Different options of symmetrization of the effective stress tensor for the unidirectionally damaged material with parallel microcracks and for the bidirectionally damaged material with a system of orthogonal microcracks were compared. The differences in any forms of the effective stress tensor were second-order infinitesimals for low damage levels. The differences in predictions from considered symmetrization schemes increased with the growth of the damage level and with that of the differences between the eigenvalues of damage. An identification procedure for anisotropic damage was proposed on the basis of acoustic emission methods and then it was discussed. Key words: continuum damage mechanics; damage tensor; effective stress tensor; anisotropy; simulation; identification

Introduction

The effective stress tensor (EST) is one of the central concepts in continuous

M

damage mechanics (CDM). It was introduced for the one-dimensional case in the

ED

first papers on the mechanics of a damaged continuum by Kachanov [1] and Rabotnov [2]. The emergence and evolution of microdefects (microcracks and

PT

micropores) leads to a reduction in the effective area of the bearing elements of the quasicontinuum, which in turn leads to an increase in internal stresses (called

CE

effective), which at the macrolevel can be regarded as averaging of the real stresses acting in the material at the microlevel. Therefore, the formulations of the

AC

phenomenological constitutive equations used within the CDM are, as a rule, related to substituting the traditional Cauchy tensor by the effective stress tensor. Noteworthy Russian and foreign reviews carried out using the CDM include

Refs. [3–7], reflecting the current state of the problem and offering a comprehensive review of all the key issues. The potential of CDM was illustrated with the analysis of creep [5–10], plasticity [6, 11, 12], fatigue [13–17, 6], combination of fatigue and creep [6, 18], thermal fatigue [18] and brittle fracture [19, 20, 6, 7] processes. 1

ACCEPTED MANUSCRIPT

Depending on the nature of the spatial and orientational distribution of microdefects, scalar [1, 2, 8, 21, etc.], vector [22– 24] and tensor (of second [8, 25– 28, etc.], fourth [29–31] and eighth [32] orders) damage parameters. There are also various approaches to determining the damage tensor: based on spectral decomposition in a basis of eigenvectors, with the principal values given as relative areas of the defects in the corresponding directions [26];

CR IP T

based on the analysis of the morphology of microcracks [25]; as the coefficient of decomposition in a generalized Fourier series for the orientational distribution of the density of microdefects [5, 28];

by introducing an equivalent loaded fictitious undamaged configuration and analyzing the corresponding deformation gradient [28] (geometric approach).

AN US

It should be noted that using the second-order damage tensor only allows us to describe the behavior of orthotropic materials, but not of the more complex symmetry groups. However, the orthotropic approximation proves to be sufficient for the case that is widespread in practice, when the damage is caused by

M

microcracking whose orientational properties are determined by the direction in

ED

which the maximum tensile stresses are applied. In the general case, the EST corresponding to the second-order damage tensor

PT

turns out to be asymmetric [26]. Handling the asymmetric stress tensor correctly leads to the need to introduce Cosserat and/or micropolar continua models. In order

CE

to avoid significantly complicating the model, it was proposed in a number of papers to symmetrize the resulting asymmetric tensor. Different versions for

AC

symmetrizing the EST have been proposed in the literature [26, 33, 34]. The purpose of this study is a comparative analysis of various versions for

symmetrizing the EST and the subsequent search for a single one-parameter formulation containing the known definitions as particular cases. Effective stress tensor Isotropic damage. In the simplest case, under uniaxial tension of a rectilinear rod with the transverse area S, whose part SD is occupied by microdefects, by the 2

ACCEPTED MANUSCRIPT

force F, the effective stress (acting in the undamaged material outside the cavities) is determined from the condition F   S   S   (S  SD ) .

If we introduce the damage as a fraction of the cross-sectional area of the rod occupied by defects D  SD / S , then the effective stress   F / S will be expressed

CR IP T

through the traditional concept of stress   F / S by the equality    / (1  D) .

In the general case under a multiaxial stressed state of an isotropic damaged material, a scalar damping parameter D (0  D  1) is also used. The EST is then

AN US

determined by the following expression: σ

1 σ. 1 D

(1)

The damage in this case is the surface density of microdefects and is determined

M

by the equality

D   SD /  S ,

ED

where δSD is the area of defects in an arbitrarily oriented surface element δS. In the absence of damage (D = 0) the EST σ is identical to the Cauchy stress tensor σ.

PT

Accepting the hypothesis of equivalent stresses ( ε  ε ), the elastic moduli tensor of the damaged continuum 4 Ce (for whose elastic state the expression

CE

σ  4 Ce  ε is valid) is related to the tensor of initial elastic moduli (undamaged

AC

material) 4 Ce (for which σ  4 Ce  ε ) by the equality 4

Ce  (1  D) 4 Ce .

(2)

An alternative representation can be obtained if the hypothesis of equivalent

(elastic) energies is accepted; this representation has the form 4

Ce  (1  D)2 4 Ce .

(3)

Anisotropic damage. An oriented surface element nδS in the actual configuration is transformed into an equivalent loaded pseudo-undamaged 3

ACCEPTED MANUSCRIPT

configuration to n S . Similar to the one-dimensional case considered above, the symmetric damage tensor D can be introduced as an element of the operator of linear transformation of nδS to n S :

1  D  n S  n S , where 1 is the unit tensor.

CR IP T

Representing damage as a second-rank tensor allows to take into account the anisotropy of damage accumulation (the emergence and growth of microcracks perpendicular to the direction in which the maximum principal stresses are applied). Imposing the condition that the resulting vectors of the ‘traditional’ and

AN US

effective stresses must coincide, we arrive at the relation σ  n S  σ  n S ,

and from here the EST can be determined [9]:

σ  σ  1  D  . 1

(4)

M

In the general case, the EST introduced by definition (4) turns out to be asymmetric. Symmetry is observed for isotropic damage D  D1 , as well as when

ED

the stress tensor σ and the damage tensor D are coaxial.

PT

In the general case, not limited to considering only the second-rank damage tensors, the relationship between the effective and traditional stresses is established

CE

by means of the fourth-rank damage effect tensor 4 M and is a linear transformation of the general form [32]: σ 4 M  σ .

AC

(5)

For the considered cases of isotropic (1) and anisotropic (4) damages, the effect

tensors are determined by the following expressions: 4

4

M





1 11  11 , 2 1  D 

(6)

1 1 1 1 1  D   1 1  D   ,  2

(7)

M

using symbols of indirect dyadic multiplication 4

ACCEPTED MANUSCRIPT

 A  B

ijkl

 Aik B jl ,  A  B ijkl  Ail Bjk .

Two terms appear in the right-hand sides of expressions (4) and (5) due to the symmetry of the tensor σ  σS 

where ...

and ...

S

T

1 2

σ  σ  , T

are the symmetrization and transposition operations

CR IP T

(permutation of indices) of a second-rank tensor. For example, in obtaining expression (4), as a consequence of formula (1), we have:

1  D  σ  σ  1  σ  1  1  12  σ  σT   1  ei  e j 12  ik km mj   ik mk mj   ei  e j 12  ik mj   im kj  km 





 12 11  11  σ ,

AN US

where  ik is the Kronecker delta,  is the symbol of direct dyadic multiplication

 A  B ijkl  Aij Bkl .

Using the hypothesis of equivalent stresses, the tensor of the elastic moduli of

M

the damaged continuum 4 Ce is related to the tensor of the initial elastic moduli by

4

ED

the equality

Ce  4 M 1  4 Ce .

(8)

AC

CE

equality

PT

In the case when the hypothesis of equivalent energies is used, we obtain the 4

Ce  4 M 1  4 Ce  4 M T .

(9)

Symmetrization of the effective stress tensor

Explicit schemes. The simplest version for symmetrizing the EST was

proposed by Murakami and Ohno [26] and consists in calculating the symmetric part of the second-rank tensor appearing in the right-hand side of equality (4): σ

1 1 1 1  D   σ  σ  1  D   .   2

(10)

The damage effect tensor can be obtained as a direct consequence of the tensor formula (10) in the form 5

ACCEPTED MANUSCRIPT 4

M

1 1 1 1 1 1 1  D   1 1  D   1  D  1  1  D  1 .  4

(11)

In the principal axes n1, n2 , n3 of the damage tensor D  D1n1  n1  D2n2  n2  D3n3  n3 ,

the matrix corresponding to the damage effect tensor (11) has the form: 0

0

0

0

M 22

0

0

0

0

M 33

0

0

0

0

M 44

0

0

0

0

0

0

0  0  0  , 0  0   M 66 

CR IP T

 M 11  0   0 [M ]    0  0   0

0

M 55 0

(12)

AN US

where the diagonal elements relating the normal and tangential stress components are determined by the expressions with different structures: M 11 

1 , 1  D1

(13)

M

1 1 1  M 44    . 2  1  D1 1  D2 

The matrix elements M22 and M33 are calculated similarly to M11. The elements M55

PT

2→3→1….

ED

and M66 are found similar to M44, based on the cyclic permutation of the indices 1 → The matrix (12) is diagonal, due to the choice of the coordinate system

CE

associated with the main axes D. In the general case, for an arbitrary coordinate system, this matrix will be different from the diagonal one.

AC

In obtaining the matrix [M] (12) the order of the indices corresponding to the

following order of enumeration of the stress tensor components was adopted:

11 , 22 , 33 ,12 , 23 , 31 .

An alternative version of symmetrization (4) was proposed in Ref. [33]: σ  1  D 

1/2

 σ  1  D 

1/2

.

(14)

If the stress tensor and the damage tensor are coaxial, the definitions (10) and (14) coincide. 6

ACCEPTED MANUSCRIPT

The effect tensor corresponding to the multiplicative symmetrization procedure (14) has the form: 4

M

1 1/2 1/2 1/2 1/2 1  D   1  D   1  D   1  D   .  2

(15)

The matrix corresponding to the effect tensor (15) can also be represented as a diagonal matrix (12) in the principal axes of the damage tensor; the components of the

M 11 

1 , 1  D1

M 44 

1

CR IP T

matrix are calculated from the expressions:

1  D1 1  D2 

(16)

.

AN US

As in the previous case, the matrix elements M22 and M33 are calculated similarly to M11, and the elements M55 and M66 similarly to M44, based on the cyclic permutation of the indices 1 → 2 → 3 → 1 …. It should be noted that the expressions for M11 are the same in (13) and (16).

M

For the sake of convenience, from now on the transformation associated with equality (10) will be called additive symmetrization, and the one associated with

ED

with equality (14) will be called multiplicative symmetrization. In this paper, it is proposed to use a unified universal representation for a

PT

symmetrized EST containing variants (10) and (14) as particular cases. The proposed

σ

1 a  (1 a )  (1 a ) a  1  D   σ  1  D   , 1  D   σ  1  D    2

(17)

AC

CE

representation is as follows:

where a is the dimensionless scalar parameter varying in the range from 0 to 1. We obtain the additive symmetrization procedure (10) for a = 0 or 1, and the multiplicative symmetrization procedure (14) for a = 1/2. The effect tensor corresponding to the generalized symmetrization procedure (17) has the form 4

M

1 a  (1 a ) a  (1 a )  (1 a ) a  (1 a ) a  1  D   1  D   1  D    1  D   1  D   1  D   1  D   1  D    4 7

ACCEPTED MANUSCRIPT

.

(18) The matrix for the effect tensor (18) is determined by the diagonal matrix (12)

in the principal axes of the damage tensor, with M11 

1 , 1  D1

(19)

CR IP T

 1 1 1 M 44    . 2  1  D1 a 1  D2 1a 1  D1 1a 1  D2 a 

M22 and M33 are calculated similarly to M11, while M55 and M66 are calculated similarly to M44, by cyclic permutation of the indices 1 → 2 → 3 → 1 … . As a further generalization of representation (17), we can consider the following

σ

AN US

two-parameter representation:

1 a b b a 1  D  σ  1  D   1  D   σ  1  D   .  2

(20)

It contains expression (17) for b = 1 – a as a particular case. The symmetrization procedure (20) makes it possible to take into account the EST

M

variant proposed by Betten in Ref. [27], having the form σ  1  D   σ  1  D  , 1

ED

1

as well as independence in choosing the stress exponents for σ and the continuity

PT

tensor Ψ  1  D in generalized formulations of the constitutive creep equations [8]. The effect tensor corresponding to the two-parameter symmetrization procedure

CE

(20) is determined by the following equation:

1 a b a b b a b a 1  D   1  D   1  D   1  D   1  D   1  D   1  D   1  D    4

AC 4

M

.

(21)

Implicit schemes. Refs. [33, 34] consider the possibility of setting the EST implicitly by solving the following equation: σ

1 1  D   σ  σ  1  D   . 2

(22)

The conditions for determining the symmetric EST obtained by implicit additive 8

ACCEPTED MANUSCRIPT

symmetrization (22) are discussed in [34]. Introducing the assumption of the symmetry of the EST, which is expressed as σ  σS 

1 2

σ  σ  , T

the effect tensor is determined in the following form: 4



 



1

1 1  M   11  11  1D  1D  D1  D1  . 4 2 

(23)

principal axes of the damage tensor, where M 11 

1 , 1  D1

1 . D1  D2 1 2

(24)

AN US

M 44  M 44 

CR IP T

The matrix corresponding to the effect tensor (23) has the form (12) in the

As in the previous cases, the matrix elements M22 and M33 are calculated similarly to M11, and the elements M55 and M66 are similar to M44, by cyclic permutation of the indices 1 → 2 → 3 → 1 … .

M

Implicit multiplicative symmetrization is determined by the expression

ED

σ  1  D   σ  1  D  1/2

1/2

.

(25)

Obviously, this definition does not differ from explicit multiplicative

PT

symmetrization (14).

CE

As a generalization of implicit symmetrization schemes (22) and (25), the one-

AC

parameter representation of the following form can be considered: σ

1 a 1 a 1 a a 1  D   σ   1  D    1  D   σ  1  D   ,   2

(26)

where a is the dimensionless scalar parameter varying in the range from 0 to 1 (the same as for representation (17). If a  0 (or a  1 ), we obtain the additive symmetrization procedure (22), if a  1/ 2 , we obtain the multiplicative symmetrization procedure (25).

The effect tensor corresponding to the generalized symmetrization procedure (26) has the form: 9

ACCEPTED MANUSCRIPT

4

M  4 1  D  1  D  a

1a

 1  D  1  D

1a

a

.

 1  D

1a

 1  D  1  D 

1a

a

a  1  D   

1

(27) The effect matrix written in the principal axes of the damage tensor has the form

(12), where 1 , 1  D1

M 44  2 1  D1  1  D2   a

1 a

 1  D1 

1 a

1  D2 

1

(28)

 . 

CR IP T

M11 

a

As in the previous cases, M 22 and M33 are calculated similarly to M11 , while M55 and M66 are found similarly to M 44 , by cyclic permutation of the indices 1 → 2 →

AN US

3→1….

It is possible to perform the inversion present in tensor (27) when calculating the elements of matrix (28) explicitly due to the diagonal form of effect matrix (12)

M

in the principal axes of the damage tensor.

Comparative analysis of different symmetrization versions

ED

The results obtained using various explicit (see formulae (10), (14), (17)) and implicit (formulae (22), (25), (26)) symmetrization procedures were compared for

PT

two idealized versions of the damaged material with different microstructures:

CE

a system of identically oriented (parallel) microcracks (vertical or inclined, see Fig. 1a and 1b), periodically repeating in two directions;

AC

a system of orthogonal microcracks with different lengths in each of the

directions (see Fig. 1c), periodic in two directions. The material is regarded as homogeneous at the macrolevel. The information

on the microstructure at the macrolevel is taken into account only by introducing the damage tensor. The tensor nature of the damage allows taking into account the anisotropy of the properties of an initially isotropic material.

10

ACCEPTED MANUSCRIPT

b)

c)

CR IP T

a)

Fig. 1. Idealized versions of the damaged material:

systems of vertical (a), unidirectional, inclined (b) and orthogonal (c) microcracks

System of parallel microcracks. In the case under consideration (see Fig.

AN US

1b), the damage tensor has the form D  D1n1  n1 ,

(29)

where D1 is the maximum eigenvalue of the damage tensor (0  D1  1) , determined by the ratio between the length of the microcrack and the size of the undamaged

M

arch (the distance between the nearest vertices of adjacent collinear cracks); n1 is the normal to the microcrack edges.

ED

The simplest case of a system of vertical microcracks (Fig. 1a, where   0 ,

PT

D  D1i  i ), allowing to obtain a visual interpretation of the results, is of particular

interest. When a macrosample is uniaxially stretched in a direction perpendicular

CE

to the orientation of the microcracks, a uniaxial homogeneous stress state will be emerge in it at the macrolevel; the stress tensor is expressed as

AC

σ  11i  i .

This tensor is coaxial with the damage tensor. In this case, the predictions for

all additive and multiplicative explicit and implicit procedures for EST symmetrization considered above coincide:  11 

 11 1  D1

.

(30)

This obvious result could also be obtained from comparing the effect matrices [M] for the considered versions of determining the EST. All matrices (13), (16), 11

ACCEPTED MANUSCRIPT

(19), (24) and (28) have the same first diagonal elements M11 in the principal axes of the damage tensor. A similar statement is also true for the second and third diagonal elements. The only differences are observed in the fourth, fifth and sixth diagonal elements of the effect matrices. This indicates that the differences may occur under shearing. При плоском чистом сдвиге, тензор напряжений которого имеет вид A

CR IP T

difference in EST predictions is observed under pure in-plane shear, where the stress tensor has the form

σ  12  i  j  j  i  .

Table 1 lists the formulae for calculating the ratios 12 / 12 , obtained as the

AN US

values of M 44 with D1  0 and D2  0 , for all explicit and implicit symmetrization procedures considered above. The relations obtained were used to compare different symmetrization schemes; the results of the comparison are shown in Fig. 2.

M

The greatest absolute values of the ratio 12 / 12 are observed for the explicit additive symmetrization scheme (10) (see Fig. 2a) over the whole variation range

ED

of the damage. The smallest values of 12 / 12 correspond to the implicit additive symmetrization scheme (22). The curves corresponding to the multiplicative

PT

symmetrization (the explicit (14) and implicit (25) schemes produce the same

CE

result in this case) occupy an intermediate position (Fig. 2a). Unlike the explicit scheme, the implicit additive symmetrization scheme does

AC

not exhibit a 12 / 12 singularity as D1 tends to unity. Table 1 The 12 / 12 ratios under pure in-plane shear in the system of vertical microcracks

EST symmetrization procedure Basic formula

additive

(10)

12 / 12 1 1  1   2  1  D1 

Curve no. in Fig. 2

1 12

ACCEPTED MANUSCRIPT

multiplicative

(14) (17)

generalized

Implicit

additive

(22)

multiplicative

(25)

generalized

(26)

2

 1 1 1   a 1 a  2  1  D1  1  D1  

4

1 1  D1 / 2 1 1  D1

3

2

1  D1 

a

3

b)

a)

4

3 2

1212

2

2

3

0,25

0,50

1 a

5

0,75

5

1

M

1 0,00

 1  D1 

AN US

1212

1

–

CR IP T

Explicit

1 1  D1

1,00

D1

0

0,0

0,2

0,4

0,6

0,8

1,0

a

PT

ED

Fig. 2. Dependence of the 12 / 12 value on the damage D1 (a) and the parameter a (for the value of D1 = 0.8) (b) in the system of vertical microcracks for different EST symmetrization schemes (see Table 1)

CE

It should be noted that comparing the second EST invariants for the considered symmetrization schemes with the second invariant of the asymmetric

AC

EST (4) indicated that the explicit additive scheme (10) yields the prediction closest to (4). The results of the analysis of the effect that the parameter a has on the 12 / 12

ratio for a fixed value of D1 = 0.8 are shown in Fig. 2b. The greatest differences in the 12 / 12 values in the explicit (17) and the implicit (26) EST symmetrization schemes are observed with a = 0 and a = 1 (corresponding to the case of additive symmetrization) and amount to 45%. The difference between the additive and multiplicative procedures of explicit symmetrization is 25%. As the damage level D1 13

ACCEPTED MANUSCRIPT

increases, these differences also increase (see Fig. 2a). Importantly, the 12 / 12 ratio is not always less for explicit than for implicit symmetrization. In contrast to the case of vertical microcracks (see Fig. 1a,), the analysis of a material with inclined microcracks (Fig. 1b,   0 ) revealed that under uniaxial tension of the representative volume, the heater will contain not one but three nonzero components, with different forecasts of the symmetrization procedures for

CR IP T

each of them. The results of calculating the 11 / 11 ratio under uniaxial loading, when

σ  11i  i , D  D1n1  n1 ,

for the considered explicit (17) and implicit (26) symmetrization schemes are

AN US

determined, respectively, by the following equations:

(31)

   11 1 4  cos 4   sin 4   sin 2  cos 2   . a 1 a   11 1  D1  1  D1   1  D1  

(32)

M

   11 1 1 1  cos 4   sin 4   sin 2  cos 2    , a 1 a   11 1  D1  1  D1  1  D1  

ED

A limit analysis of the relations (31) and (32) with D1 1 has shown that the differences for any EST variants are of the second order of smallness O  D12  .

PT

Fig. 3 shows a comparison of the 11 / 11 values obtained by means of the explicit additive, explicit multiplicative and implicit additive symmetrization

CE

procedures with different values of the orientation angle φ of the microcracks and a

AC

constant value of D1 = 0.8. The maximum 11 / 11 value (as in the aboveconsidered case (see Fig. 2a)) is observed for the explicit additive symmetrization scheme (10); the smallest 11 / 11 value corresponds to the implicit additive symmetrization scheme (22); multiplicative symmetrization (the explicit and the implicit schemes yield the same result) occupies an intermediate position (see Fig. 3). The maximum differences in the 11 / 11 values for the explicit and the implicit symmetrization schemes are observed with   45 .

14

ACCEPTED MANUSCRIPT

90 5

120

60

4 3

1 2

150

30

1111

2



3

1 0 180

0

1

3

CR IP T

2 210

330

4 240

5

300

270

AN US

Fig. 3. Dependences of the 11 / 11 ratio on the orientation angle φ of inclined microcracks for three symmetrization schemes of the effective stress tensor: explicit additive scheme (formula (10), curve 1); explicit multiplicative scheme (formula (14), curve 2), implicit additive scheme (formula (22), curve 3)

The nature of the 11 / 11 ratio variation with the growth of the damage D1 for a

M

material with inclined microcracks generally corresponds to that observed for the 12 / 12 ratio (compare Figs. 4a and 2a). However, the differences between the

ED

predictions for different EST symmetrization schemes in the case under consideration (   45 ) are less substantial. In contrast to 12 / 12 , the implicit additive

PT

symmetrization scheme for 11 / 11 exhibits a singular behavior for 11 / 11 as D1

8

3

b)

AC

a)

CE

tends to unity.

4 5 2

1111

1111

6

2

4

1

3

1

2

0 0,0

0,2

0,4

0,6

0,8

1,0

D1

0 0,0

0,2

0,4

0,6

0,8

1,0

a

15

ACCEPTED MANUSCRIPT

Fig. 4. Dependences of the 11 / 11 value on the damage D1 (a) and the parameter a (with the value of D1 = 0.8) (b) in the system of inclined microcracks (φ = 45º) for different symmetrization schemes of the effective stress tensor: explicit additive scheme (formula (10), curve 1); explicit multiplicative scheme (formula (14), curve 2), implicit additive scheme (formula (22), curve 3), explicit scheme (formula (31), curve 4), implicit scheme (formula (32), curve 5)

Fig. 4b shows the results of the analysis of the effect that the parameter a has

CR IP T

on the 11 / 11 ratio with fixed values of the parameters D1 = 0.8 and φ = 45º for the system of inclined cracks. The same as in the case of vertical microcracks for 12 / 12 (see Fig. 2b), the greatest differences in the 11 / 11 values for the explicit

(31) and the implicit (32) symmetrization schemes are observed with a = 0 and a =

AN US

1 (corresponding to the case of additive EST symmetrization). The 11 / 11 ratio for explicit EST symmetrization is always not less than for implicit symmetrization. The system of orthogonal microcracks. The damage tensor in the case of the system of orthogonal microcracks (see Fig. 1c) has the form

M

D  D1n1  n1  D2n2  n2 .

(33)

ED

The values of D1 and D2 are determined by the ratio between the lengths of the cracks and the gaps between them (the distances between the vertices of

PT

adjacent collinear cracks) in the directions n1 and n2 (the principal axes of the damage tensor), respectively. The components of the damage tensor D1 and D2

CE

satisfy the constraints

0  D1  1 ; 0  D2  1.

AC

The results of calculating the 11 / 11 ratio for a material damaged by a system

of orthogonal microcracks (see Fig. 1c) with the tensor D determined by formula (33) under uniaxial loading, with the stress tensor expressed as σ  11i  i ,

for the previously considered explicit (17) and implicit (26) symmetrization procedures are determined, respectively, by the following equations:

16

ACCEPTED MANUSCRIPT    11 cos 4  sin 4  1 1    sin 2  cos 2    , a 1 a a 1 a   11 1  D1 1  D2 1  D2  1  D1    1  D1  1  D2 

(34)

   11 cos 4  sin 4  4    sin 2  cos 2   . a 1 a a 1 a   11 1  D1 1  D2  1  D1  1  D2   1  D2  1  D1  

(35)

The only difference between Eqs. (34) and (35) is in the expressions in square

Using the inequality 1 1 4 ,   x y x y

CR IP T

brackets.

valid for x  0, y  0 , it is easy to show that the value of the 11 / 11 ratio calculated

AN US

by formula (34) is always greater than or equal to the corresponding value found by formula (35). The equalities (31) and (32) introduced above enter as particular cases into formulae (34) and (35) with D2 = 0.

A comparison of the 11 / 11 values found using formulae (34) and (35)

M

showed that the maximum 11 / 11 ratio is observed for all values of the angle φ for the explicit additive symmetrization scheme (10) (particular case (34) with a = 0);

ED

the minimum 11 / 11 value corresponds to the implicit additive symmetrization scheme (22) (particular case (35) with a = 0); multiplicative symmetrization (the

PT

explicit and implicit schemes yield the same result) occupies an intermediate

CE

position. The maximum differences in the 11 / 11 values for the explicit and implicit EST symmetrization schemes are observed with   45 .

AC

The nature of the 11 / 11 variation with increasing damages D1 and D2

corresponds to that observed for unidirectionally oriented microcracks (see Fig. 4, a). The dependence for the explicit additive EST symmetrization scheme (10) is shown in Fig. 5.

17

ACCEPTED MANUSCRIPT 5

11

4

11 

3 2 1 0 ,8

D 2

CR IP T

0 , 80

0 ,6

0 ,6

0 ,4

0 ,4 0 ,2

0 ,2 0 ,0

0,0

D1

AN US

Fig. 5. Dependence of the 11 / 11 ratio on the damage parameters D1 and D2 in the system of orthogonal microcracks with   45 for explicit additive symmetrization (see formula (10)) The effect of D2 on the 11 / 11 angular distribution with a fixed D1 value is shown in Fig. 6 for the explicit additive EST symmetrization scheme (10). As the

M

D2 value tends to D1, the 11 / 11 curve tends to a circle corresponding to the case of

PT

ED

isotropic damage.

5

90 120

1

1111

AC

CE

4 3

60

150

30

2

2



3

1 0 180

0

1 2 3

210

330

4 5

240

300 270

Fig. 6. Dependences of the 11 / 11 value on the orientation angle of microcracks for different ratios of D1 and D2 with a fixed value of D1 = 0.8. The values of D2 = 0.7 (curve 1), 0.5 (2), 0.0 (3) 18

ACCEPTED MANUSCRIPT

The 11 / 11 isolines on the D1–D2 plane (Fig. 7), constructed using different EST symmetrization schemes, show that the difference in the predictions increases with an increase in the difference between D1 and D2. The predictions are identical for D1 = D2. The greatest discrepancy is observed for unidirectionally oriented microcracks with either D1 or D2 equal to zero (see Fig. 7). The spread in the orientation angles of the microcracks reduces the discrepancy in the predictions of the EST symmetrization

D2

1111=3.0

0,8

0,4

2.5

3

AN US

0,6

2.0

12

1.5

M

0,2

0,2

0,4

0,6

0,8

D1

ED

0,0 0,0

CR IP T

schemes.

PT

Fig. 7. The isolines of the 11 / 11 values on the D1 – D2 plane for different EST symmetrization schemes:

CE

explicit additive, formula (10) (curve 1); explicit multiplicative, formula (14) ( curve 2); implicit additive, formula (22) (curve 3)

The effect of the scalar parameter a on the 11 / 11 ratio for a material damaged

AC

by a system of orthogonal microcracks has a character similar to the previously considered case of unidirectionally oriented microcracks (see Fig. 4b). The values of the 11 / 11 ratio for explicit EST symmetrization are always greater than or equal to the corresponding values for implicit symmetrization with any admissible values of D1, D2 and φ, which follows directly from the analysis of the form of relations (34) and (35). Identification of the components of the damage tensor 19

ACCEPTED MANUSCRIPT

The known methods of damage assessment are based on measuring physical quantities [35]: the moduli of elasticity and density of the material, its microhardness, the velocity of propagation of ultrasonic waves in the material, the deformations of the accelerated creep stage, the electrical resistance of the material, etc. Acoustic emission methods based on measuring the anisotropy of the acoustoelastic properties of a solid show promise for analysis of anisotropic damage

CR IP T

[36, 37]. The phase velocities v of the propagation of a plane monochromatic wave u  A expik  N  r  vt 

in the direction N in an elastic medium characterized by a tensor of elastic moduli 4

Ce are determined on the basis of the solution of the equation

AN US

det  N  4 Ce  N   v 2   0 ,

which is a condition for determining the eigenvalues of the acoustic tensor N  4 Ce  N .

M

Above we have introduced the notation ρ for the density of the material. Using the equivalent strain hypothesis (8), the equation for determining the

PT

moduli tensor

ED

propagation velocities v of the waves in a damaged continuum with the elastic

4

Ce  4 M 1  4 C

CE

has the form

det  N 4  4 M 1  4 Ce  N   v 2   0 .

(36)

AC

The solution of Eq. (36) takes a cumbersome form in the general case. It can

be simplified by considering the propagation of a wave in the direction of one of the principal axes of the damage tensor, for example, along the third axis N  n3 . In this case, for an initially isotropic material or for an initially orthotropic material (provided that the anisotropy axes coincide with the principal axes of the damage tensor), the acoustic tensor is diagonal and it is possible to construct the solution of Eq. (36) in a simple compact form: 20

ACCEPTED MANUSCRIPT v  M 1C e  1 33 33  3  1 e 1 v2  M 55 C55   1 e 1 v1  M 66 C66 

(37)

where v3 is the propagation velocity of a longitudinal wave in the direction n3 ; v1 , v2 are the velocities of transverse waves in the directions n1 and n 2 , respectively.

For an initially isotropic elastic material characterized by Lamé’s constants λ

4



CR IP T

and µ and determined by the tensor of elastic moduli taking the form



Ce  1  1   11  11 ,

(38)

using the explicit symmetrization scheme (17), taking into account formula (19), as

AN US

a consequence of relations (37), we obtain the following expressions for the velocities:

 

(39)

ED

 

M

 1/2 v3  1  D3    2   /    1/2  a a 1 a 1  a 1   . v  1  D 1  D  1  D 1  D 2  /      2  3 2 3 2   1/2  a a 1 a 1  a 1 v1  1  D3  1  D1   1  D3  1  D1   2  /  

Using the implicit symmetrization scheme (26), taking into account formula

1/2  v3  1  D3    2   /    1/2 a 1 a 1 a a v2  1  D3  1  D2   1  D3  1  D2   / 2   /   v   1  D a 1  D 1 a  1  D 1 a 1  D a  / 2   /  1/2   1  3  1 3  1 

AC

CE

velocities:

PT

(28), as a consequence of relations (37), we obtain alternative expressions for the

 

 

(40)

Expressions (39) or (40) for the velocities v1 , v2 , v3 can be regarded as a system

of three nonlinear algebraic equations with respect to three unknown principal values of the damage tensor D1, D2 , D3 . In a number of cases, the solution can be obtained analytically in closed form.

21

ACCEPTED MANUSCRIPT

In the case of explicit additive symmetrization, the principal damages will be found as the solutions of system (39) with a = 0: 2   D3  1   v3 / v3   2 2 1  D2  1   2  v2 / v2    v3 / v3   ,  1  D  1   2  v / v 2   v / v 2  3 3  1  1 1 

(41)

CR IP T

where v3     2  /  is the propagation velocity of a longitudinal wave in the undamaged material, v1  v2   /  are the propagation velocities of transverse waves in this material.

Expressions (41) can be directly used to identify anisotropic damage based on

AN US

velocity measurements of ultrasonic signals [36–38].

In the case of explicit multiplicative symmetrization, as a consequence of system of equations (39), with a = 1/2, we have:

(42)

M

 D3  1   v3 / v3 2  4 2   D2  1   v2 / v2  /  v3 / v3   4 2  D1  1   v1 / v1  /  v3 / v3 

ED

In the case of implicit additive symmetrization, the damages will be determined by searching for the solution of the system of equations (40), with a =

AC

CE

PT

0:

2   D3  1   v3 / v3  2 2   D2  1   2  v2 / v2    v3 / v3   .   D1  1   2  v1 / v1 2   v3 / v3 2    

(43)

Similar expressions for the damages can be obtained if the equivalent energy

hypothesis (9) is used. The wave propagation velocities assuming that the damage is constant are determined in this case as the solutions of the characteristic equation: det  N 4  4 M 1  4 Ce  4 M T  N   v 2   0 .

(44)

22

ACCEPTED MANUSCRIPT

The principal values of the damage tensor using Eq. (44) is carried out are identified similar to the case of analysis (36) considered above. For explicit additive symmetrization (10), we have:  D3  1  v3 / v3  1   D2  1   2  v2 / v2    v3 / v3   ;  1  D1  1   2  v1 / v1    v3 / v3  

(45)

CR IP T

in case of explicit multiplicative symmetrization (14), we obtain the expressions:  D3  1  v3 / v3  2  D2  1   v2 / v2  /  v3 / v3  ;  2  D1  1   v1 / v1  /  v3 / v3 

(46)

AN US

in case of implicit additive symmetrization (22), we arrive at the equations:  D3  1  v3 / v3   D2  1   2  v2 / v2    v3 / v3   .   D1  1   2  v1 / v1    v3 / v3  

(47)

M

The arbitrariness in choosing the EST symmetrization scheme and the equivalence hypothesis can be eliminated by analyzing the experimental data for a

ED

particular type of material and loading conditions. It is necessary to ensure that the

PT

inequalities

0  D1  1 , 0  D2  1, 0  D3  1

CE

are satisfied, that the damage is increased monotonously with increasing load, that it is possible to obtain a thermodynamically consistent and simplest form of the

AC

evolution equations for the damage tensor.

Conclusion

The proposed unified generalized one-parameter form of the representation for the

symmetrized EST allows to introduce new types of the EST, as well as to analyze the properties of the damaged continuum in a general form for different ESTs. . It was established in the present paper by solving a number of test problems for parallel and orthogonal microcracks that the explicit additive symmetrization scheme 23

ACCEPTED MANUSCRIPT

yields the upper bound of the effective stress tensor (i.e., it is a conservative estimate), while the implicit additive scheme yields the lower bound. The differences in the predictions of the considered symmetrization schemes increase with an increase in the damage level, as well as with increasing differences between the principal values of the damage tensor. At low levels of damage, the differences in any versions of the EST are second-order infinitesimals.

CR IP T

The variability in determining the symmetrized EST leads to different equivalent definitions of damage. In a number of cases, choosing the most suitable version of the EST can be solved on the basis of experimental studies by identifying the parameters of the equation of damage evolution. The paper also proposes a procedure for identifying

AN US

anisotropic damage based on the widely known acoustic methods.

The paper was financially supported by a Russian Science Foundation grant (project no. 15-19-00091).

REFERENCES

M

[1] L.M. Kachanov, O vremeni razrusheniya v usloviyakh polzuchesti [On the creep-rapture life], Izv. AN SSSR. Ser. Techn. Sci. (8) (1958) 26–31.

ED

[2] Yu.N. Rabotnov, O mekhanizme dlitelnogo razrusheniya [On the delayed fracture mechanics], Voprosy prochnosti materialov i konstruktsiy [On the strength

PT

of materials and constructions], Moscow, AN SSSR (1959) 5–7.

CE

[3] P.S. Volegov, D.S. Gribov, P.V. Trusov, Povrezhdennost i razrusheniye: klassicheskiye kontinualnyye teorii [The damage and fracture: Classical continuum

AC

theories], Fizicheskaya mezomekhanika. 18 (4) (2015) 68–86. [4] A.M. Lokoshchenko, Primeneniye kineticheskoy teorii pri analize

dlitelnogo vysokotemperaturnogo razrusheniya metallov v usloviyakh slozhnogo napryazhennogo sostoyaniya. Obzor [An application of the kinetic theory in the analysis of the delayed high-temperature fracture of metal under complex stress condition (Review)], Priklad. Mekhanika i Tekhnich. Fizika. 53 (4) (2012) 149– 164.

24

ACCEPTED MANUSCRIPT

[5] S. Murakami, Continuum damage mechanics, Springer, Heidelberg, London, New York (2012). [6] J. Lemaitre, R. Desmorat, Engineering damage mechanics: ductile, creep, fatigue and brittle failures, Springer, Berlin (2005). [7] D. Krajcinovic, Damage mechanics, Elsevier, Amsterdam (1996). [8] Yu. N. Rabotnov, Polzuchest elementov konstruktsiy [Creep of structural

CR IP T

elements], Moscow, Nauka (1966). [9] J. Hult, Creep in continua and structures, Topics in Applied Continuum Mechanics, Ed. by J.L. Zeman and F.Ziegler, Vienna, Springer (1974) 137–155. [10] F. Leckie, D. Hayhurst, Creep rupture of structures, Proc. R. Soc. Lond. A340 (1622) (1974) 323–347.

AN US

[11] A. Dragon, Plasticity and ductile fracture damage: study of void growth in metals, Eng. Fract. Mech. 21 (4) (1985) 875–885.

[12] J. Lemaitre, A continuous damage mechanics model for ductile fracture, J. Eng. Mater. Technol. 107 (1) (1985) 83–89.

M

[13] J. Lemaitre, Evaluation of dissipation and damage in metals under

(1971) 540–549.

ED

dynamic loading, Proc. Int. Conf. on the Mech. Behavior of Materials I. Japan,

PT

[14] J.L. Chaboche, Une loi différentielle d’endommagement de fatigue avec cumulation non linéaire, Revue Française de Mécanique. (50–51) (1974) 71–82.

CE

[15] A.S. Semenov, Issledovaniye protsessa obrazovaniya zigzagov pri rasprostranenii

ustalostnoy

treshchiny

metodami

kontinualnoy

mekhaniki

AC

povrezhdeniy [A study of zigzag formation during the fatigue crack propagation by methods of the continuum damage mechanics], Nelineynyye problemy mekhaniki i fiziki deformiruyemogo tverdogo tela, Ed. Chernykh K.F., Sbornik Trudov Nauchnoy Shkoly akademika V.V. Novozhilova, St. Petersburg (2000) 186–211. [16] A.S. Semenov, Mekhanika nelokalnykh kontinualnykh povrezhdeniy [Nonlocal continuum damage mechanics], St. Petersburg Polytechnical State University Journal. No. 1 (48) (2006) 148–159.

25

ACCEPTED MANUSCRIPT

[17] B.E. Melnikov, A.S. Semenov, Energeticheskaya model nakopleniya povrezhdeniy pri malotsiklovoy ustalosti [An energetic model of damage accumulation during low-cycle fatigue], XXI Peterburgskiye chteniya po problemam prochnosti. K 100-letiyu so dnya rozhdeniya L.M. Kachanova i Yu.N. Rabotnova. Sb. Materialov, St. Petersburg, Solo (2014) 59–61. [18] A.S. Semenov, L.B. Getsov, Kriterii termoustalostnogo razrusheniya

CR IP T

monokristallicheskikh zharoprochnykh splavov i metody opredeleniya ikh parametrov [Thermal fatigue fracture criteria of single crystal heat-resistant alloys and methods for identification of their parameters], Strength of Materials. 46 (1) (2014) 38–48.

[19] D. Krajcinovic, G.U. Fonseka, The continuous damage theory of brittle

AN US

materials, Parts I and II, ASME, J. Appl. Mech. 48 (4) (1981) 809–824.

[20] A.V. Benin, A.S. Semenov, S.G. Semenov, B.E. Melnikov, Matematicheskoye modelirovaniye protsessa razrusheniya stsepleniya armatury s betonom. Chast 2. Modeli bez ucheta nesploshnosti soyedineniya [The simulation

M

of bond fracture between reinforcing bars and concrete. Part 2. Models without

(2014) 23–40.

ED

taking the bond discontinuity into account], Magazine of Civil Engineering. 45 (1)

PT

[21] P.A. Pavlov, Osnovy inzhenernykh raschetov elementov mashin na ustalost i dlitelnuyu prochnost [A basis of design studies of machine parts for

CE

fatigue and long-term strength], Leningrad, Mashinostroyeniye (1988). [22] L.M. Kachanov, Osnovy mekhaniki razrusheniya [A basis of the failure

AC

mechanics], Moscow, Nauka (1974). [23]

Yu.K.

Petrenya,

Fiziko-mekhanicheskiye

osnovy

kontinualnoy

mekhaniki povrezhdayemosti [Physical and mechanical foundations of continuum damage mechanics], St. Petersburg, Politekhnika (1997). [24] L. Davison, A. Stevens, Thermomechanical constitution of spalling elastic bodies, J. Appl. Phys. 44 (2) (1973) 667–674.

26

ACCEPTED MANUSCRIPT

[25] A.A. Vakulenko, M.L. Kachanov, Kontinualnaya model sredy s treshchinami [Continuous model of the medium with cracks], Izv. AN SSSR. Mekh. Tverd. Tela. (4) (1971) 159–166. [26] S. Murakami, N. Ohno, A continuum theory of creep and creep damage, Creep in Structures, A.R.S. Ponter, D.R. Hayhurst (eds.), Berlin, Springer (1981) 422–443.

CR IP T

[27] J. Betten, Damage tensors in continuum mechanics, Journal de Mécanique Théorique et Appliqué. 2 (1) (1983) 13–32.

[28] Yu.N. Radayev, Tenzornyye mery povrezhdennosti i garmonicheskiy analiz tonkoy struktury povrezhdennosti

[Tensor damage measures and a

No. 2 (8) (1998) 79–105.

AN US

harmonic analysis of the damage fine structure], Vest. Samarskogo Gos. Un-ta.

[29] J. Chaboche, Continuum damage mechanics: Part I, General concepts, J. Appl. Mech. 55 (1) (1988) 59–64.

[30] D. Krajcinovic, Damage mechanics, Mech. Mater. 8 (2-3) (1989) 117–

M

197.

ED

[31] F.A. Leckie, E.T. Onat, Tensorial nature of damage measuring internal variables, Physical Non-Linearities in Structural Analysis, Springer, Berlin (1981)

PT

140–155.

[32] J.-L. Chaboche, Le concept de contrainte effective applique a l'elasticite

CE

et a la viscoplasticite en presence d'un endommagement anisotrope, Mechanical Behavior of Anisotropic Solids, Boehler J.P. (ed.). Martinus Nijhoff, Hague (1982)

AC

737–760.

[33] J.P. Cordebois, F. Sidoroff, Damage induced elastic anisotropy,

Mechanical Behavior of Anisotropic Solids, Boehler J.P. (ed.). Martinus Nijhoff, Hague (1982) 761–774. [34] J. Betten, Net-stress analysis in creep mechanics, Ing.-Archiv. 52 (6) (1982) 405–419. [35] J. Lemaitre, J. Dufailly, Damage measurements, Eng. Fract. Mech. 28 (5-6) (1987) 643–661. 27

ACCEPTED MANUSCRIPT

[36] N.E. Nikitina, Akustouprugost. Opyt prakticheskogo primeneniya [Acoustoelastisity. Practical experience], Nizhniy Novgorod, TALAM (2005) [37] A.K. Belyayev, A.M. Lobachev, V.S. Modestov, et al., Otsenka velichiny plasticheskikh deformatsiy s ispolzovaniyem akusticheskoy anizotropii [Estimating the plastic strain with the use of acoustic anisotropy], Mechanics of Solids. 51 (5) (2016) 606-611.

CR IP T

[38] A.K. Belyaev, V.A. Polyanskiy, A.M. Lobachev, et al., Propagation of sound waves in stressed elasto-plastic material, Proc. Int. Conf. Days on Diffraction. No. 7756813 (2016) 56–61.

AN US

Received 18.03.2017, accepted 09.04.2017.

THE AUTHOR

AC

CE

PT

ED

M

Artеm S. Semenov Peter the Great St. Petersburg Polytechnic University 29 Politekhnicheskaya St., St. Petersburg, 195251, Russian Federation [email protected]

28