F. D. Fischer1
Professor of Mechanics,
Mem. ASME
E. R. Oberaigner
Associate Professor,
Institute for Mechanics
Montanuniversitat Leoben,
A-8700 Leoben, Austria
1
Deformation, Stress State, and
Thermodynamic Force for a
Transforming Spherical Inclusion
in an Elastic-Plastic Material
The progress of a transformed phase into an elastic-plastic parent phase is simulated by
a growing sphere. The transformation is accompanied by a dilatational volume change.
The strain and stress state in the full space is presented. In addition, the local and global
energy terms are calculated. Finally the thermodynamic forces on the interface are derived. Also strain hardening is considered. 关S0021-8936共00兲00304-4兴
Introduction
In material physics a sphere with radius R(t), t being time, is
very often used to represent a phase transformation nucleus or a
transforming microregion embedded in a parent phase. Roitburd
and Temkin 关1,2兴 and Kaganova and Roitburd 关3兴 investigated the
martensitic transformation in an elastic-plastic material by such a
model. The transformation volume strain ␧* ⫽␧ o I, I unity tensor,
gives rise to an internal strain and stress state due to its accommodation by the surrounding material. A slightly more general
study for an elastic material with different material properties was
recently presented by Nazyrov and Freidin 关4兴.
Spheres as the product phase are also dealt with by several
researchers on diffusional transformation see, e.g., Vandermeer
关5兴.
Analytical solutions for the strain and stress state of the sphere,
imbedded in an infinite elastic-plastic material, with fixed radius R
subjected to a sudden dilatational eigenstrain are usually applied.
This problem was surprisingly often treated in an independent
way in the literature. However, a comprehensive collection of all
results is still lacking. The paper by Lee et al. 关6兴 as well as the
review article by Fischer et al. 关7兴 may be cited. Several mistakes
exist in the diverse papers; these will be avoided in the following
context. Since the pressure at the interface between the internal
sphere with radius R and the outer elastic-plastic space 共hollow
sphere兲 can be considered as the unknown quantity, the solution is
directly related to the stress and strain field of a hollow sphere
under internal pressure, for details see, e.g., Lubliner 关8兴 chapter
4.3.2.
The following aspects must be taken into account when dealing
with a growing spherical inclusion instead of a fixed spherical
inclusion:
• An incremental concept must be followed with respect to the
development of the plastic strain and the plastic work.
• Under the assumption of pure radial loading, the incremental
and deformation theories of plasticity are identical. If no unloading occurs, the solution offered in the literature for a fixed radius
R can be used, as can be seen by an argument by Budiansky 关9兴.
However, with respect to the total plastic work it must be considered that also the material inside the sphere, 0⭐r⭐R(t), has
plastified.
• The growth process can also be modeled by a sequence of
elastic hollow spheres, see Franciosi et al. 关10兴, introducing a certain scaling factor, however, with much more computational effort
than below.
The main goal of this paper is to provide the researcher in the
field of phase transformations in an elastic-plastic material with a
full and comprehensive set of relations for the strain and stress
state as well as energy terms relevant for the understanding of the
growth process.
2
Analysis
2.1 Strain, Stress, and Energy Terms. We solve now the
problem of a sphere subjected to a dilational eigenstrain and imbedded in an elastic-plastic space. Since we think of phase transformations, ␧ o is mostly on the order of some few percents, and
thus we apply the infinitesmal deformation theory.
We introduce a stress vector ␴T ⫽( ␴ r , ␴ ␸ , ␴ ␪ ), where the superscript T stands for ‘‘transposed,’’ a strain vector ␧T
⫽(␧ r ,␧ ␸ ,␧ ␪ ), the three vectors eT ⫽(1,1,1), ẽT ⫽(0,1,1), eM T
⫽(2,⫺1,⫺1), the unity matrix I 共I i j ⫽ ␦ i j , Kronecker delta兲 and
the matrix J, J i j ⫽1. The yield stress is ␴ f . Linear hardening is
described by E p , being the plastic tangent modulus, and a hardening factor f ⫽1⫹2(1⫺ ␯ )E p /E. If not explicitly mentioned an
elastic-ideally plastic material with f ⫽1 is assumed.
The strain vector is decomposed into an elastic part ␧e , a plastic part ␧p , and within 0⭐r⭐R a transformation part ␧* ⫽␧ o e.
Due to Hooke’s law ␧e can be written as
E␧ e ⫽ 关共 1⫹ ␯ 兲 I⫺ ␯ J兴 • ␴.
Since a hydrostatic load stress ␴ o does not introduce any stress
deviator component, the following results can be superposed on
the corresponding elastic solution for ␴ o .
An explicit relation for R p , the radius of the outer boundary of
the plastic zone, can be found as
冉
Rp
E␧ o
⫽
R
共 1⫺ ␯ 兲 ␴ f
1
To whom correspondence should be addressed.
Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF
MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF APPLIED
MECHANICS. Manuscript received by the ASME Applied Mechanics Division, June
23, 1999; final revision, Feb. 14, 2000. Associate Technical Editor: M.-J. Pindera.
Discussion on the paper should be addressed to the Technical Editor, Professor
Lewis T. Wheeler, Department of Mechanical Engineering, University of Houston,
Houston, TX 77204-4792, and will be accepted until four months after final publication of the paper itself in the ASME JOURNAL OF APPLIED MECHANICS.
Journal of Applied Mechanics
(1)
冊
1/3
⫽ ␬ 1/3,
(2)
see, e.g., Fischer et al. 关7兴 or Böhm et al. 关11兴.
Of course, since R p ⭓R, the limit condition for plastification
ensures that the plasticity constant is greater than or equal to one.
Three different zones can be distinguished with respect to the
stress state ␴ and the corresponding strain components 共␧e is directly calculated by applying Hooke’s law兲:
Copyright © 2000 by ASME
DECEMBER 2000, Vol. 67 Õ 793
Fig. 1 Profile of the relative radial stress 円 ␴ r Õ ␴ f 円 and the relative plastic strain
円 ␧ p , r Õ ␧ o 円 in relation to r Õ R for two values of ␬
• The inner elastic zone 0⭐r⭐R(t):
2␴ f
␴⫽⫺
共 1⫹ln ␬兲e,
3
E␧e⫽⫺共1⫺2␯兲
␧p ⫽0,
(3a)
2␴ f
共1⫹ln ␬兲e,
3
␧* ⫽␧ o e.
• The elastic-plastic zone R(t)⭐r⭐R p (t):
r
2␴ f
2␴ f
e⫺
␴⫽⫺
ln ␬⫺ln
3
3
R
冉 冊册
冋 冉 冊册
冋冉 冊 册
冋
2␴ f
r
ln ␬⫺ln
E␧e ⫽⫺ 共 1⫺2 ␯ 兲
3
R
␧p ⫽⫺
u 兩 R ⫽⫺ 共 1⫺2 ␯ 兲
R
r
3
⫺
3
• The outer elastic zone R p ⭐r⭐⬁:
␴f R
␴⫽⫺ ␬
3
r
冉冊
(3c)
• The inner elastic zone 0⭐r⭐R(t):
2共1⫺2␯兲 ␴2f
U⬘⫽
• •共1⫹ln ␬兲2.
3
E
(3d)
• The elastic-plastic zone R(t)⭐r⭐R p (t):
2共1⫺2␯兲 ␴2f
r 3 2 共1⫹␯兲 ␴2f
⫹
U⬘⫽
• • ln ␬⫺ln
.
3
E
R
3 E
共 1⫹ ␯ 兲
e⫺
␴ f eM ,
3
1
␧ eM .
␬ o
(3e)
3
eM ,
冉冊
␴f R
E␧e ⫽⫺ 共 1⫹ ␯ 兲 ␬
3
r
␧p ⫽0.
(3f)
3
e5 ,
(3g)
Figure 1 demonstrates profiles for the relative radial stress
兩 ␴ r / ␴ f 兩 and the relative radial plastic strain 兩 ␧ p,r /␧ o 兩 in relation
to the dimensionless radius r/R for two values of ␬.
The plastic strain attains a maximum value of ␧p,max⫽⫺关1
⫺1/␬ 兴 ␧ o eM at r⫽R.
It is important to note that the inner zone 0⭐r⭐R(t) which
behaves elastically, has achieved a plastic strain in the amount of
␧p ⫽⫺ 关 1⫺1/␬ 兴 ␧ o eM during the growth process from R⫽0 to its
actual value. This means that no jump of the plastic strain at the
interface r⫽R(t) can be observed, see 共3e兲.
Due to spherical symmetry the total strain components are ␧ r
⫽ ⳵ u/ ⳵ r and ␧ ␸ ⫽␧ ␪ ⫽u/r, which means that due to the compatibility ␧ ␸ and ␧ ␪ must be continuous. The condition can easily be
verified with the above relations. Therefore, the radial displacement u 兩 R can be calculated from ␧ ␸ at r⫽R, 共3b兲, as
794 Õ Vol. 67, DECEMBER 2000
册
(3h)
The strain energy density U ⬘ ⫽1/2␴T • ␧e can be calculated for
the three regions as follows:
3
e⫹ ␴ f ẽ,
冋
2 共 1⫺2 ␯ 兲
␴f
R 共 1⫺ ␯ 兲 ␬ ⫺
共 ln ␬ ⫹1 兲 .
E
3
⫽
(3b)
2␴ f
共 ln ␬ ⫹1 兲 R⫹␧ o R
3E
(4a)
冉 冊册
冋
(4b)
• The outer elastic zone R p ⭐r⭐⬁:
2
共1⫹␯兲 ␴ f 2 R 6
.
U⬘⫽
• •␬
3
E
r
冉冊
(4c)
Note the jump in the specific strain energy at r⫽R(t)!
We can now calculate the total strain energy produced during
the growth of the inclusion from the origin to a certain radius R.
Employing (4a) – (4c) yields after some integration
U⫽
冕
⬁
U ⬘ 4r 2 ␲ dr⫽
0
␴ 2f 4R 3 ␲
•
• 共 1⫺ ␯ 兲共 2 ␬ ⫺1 兲 .
E
3
(5)
The increment dW ⬘p of the specific plastic work in the plastic
zone follows as
dW ⬘p ⫽ ␴T •d ␧p ,
and with 共3c兲 and 共3e兲
再
dW ⬘p ⫽ ⫺
冋
冉 冊册
r
2␴ f
ln ␬ ⫺ln
3
R
⫽
3
eT ⫺
(6a)
冎冉
冊
␴f T
3R 2
Me • ⫺ 3 ␧ o eM dR
3
r
␴ 2f
6R 2
R2
•
␧
␴
dR⫽6
1⫺
␯
•
␬
•
dR.
兲
共
r3 o f
E
r3
(6b)
The total increment of plastic work dW p then follows as
Transactions of the ASME
Fig. 2 Plastic work W P in relation to total work W T , total work W T in relation to total work W cT in a purely elastic comparison material, dimensionless
transformation barrier F B ,int . „ E Õ ␴ f2 … as function of the plasticity constant ␬.
dW p ⫽
冕
Rp
R
dW ⬘p 4r 2 ␲ dr⫽8 共 1⫺ ␯ 兲 •
The ratios W T /W Tc
␴ 2f
• ␬ ln ␬ •R 2 ␲ dR,
E
(6c)
and total plastic work W p done during the growth of the inclusion
from the origin to a certain radius R becomes
␴ 2f 4R 3 ␲
•
• 共 1⫺ ␯ 兲共 2 ␬ ln ␬ ⫹2 ␬ ⫺1 兲 ,
E
3
(7)
with the amount of total mechanical energy W Tc which would
occur in a purely elastic material. We can use the results from
Fischer et al. 关7兴 leading to
• the inner elastic zone 0⭐r⭐R(t):
2
2 共 1⫺2 ␯ 兲
␴⫽⫺
␧ Ee, E ␧e ⫽⫺
␧ Ee, ␧* ⫽␧ o e,
3 共 1⫺ ␯ 兲 o
3 共 1⫺ ␯ 兲 o
(8a)
2 共1⫺2␯兲 2
UT⬘⫽
␧ E;
(8b)
3 共1⫺␯兲2 o
• the outer elastic zone R(t)⭐r⭐⬁:
R 3
R
1
共 1⫹ ␯ 兲
␧ oE
eM , E ␧e ⫽⫺
␧ E
␴⫽⫺
3 共 1⫺ ␯ 兲
r
3 共 1⫺ ␯ 兲 o r
冉冊
冉冊
共1⫹␯兲 2 R 6
UT⬘⫽
␧E
,
3共1⫺␯兲2 o r
⬁
␧2oE 4R3␲
WTc⫽ UT⬘ 4r2␲dr⫽
.
•
3
共1⫺␯兲
0
冕
Journal of Applied Mechanics
(9a)
W p /W T ⫽
2 ␬ ln ␬
2 ␬ ln ␬ ⫹2 ␬ ⫺1
(9b)
(6d)
Böhm et al. 关11兴 pointed out that Lee et al. 关6兴 reported a different
result, namely W̃ p ⫽ ␴ 2f /E•4R 3 ␲ /3•2(1⫺ ␯ ) ␬ 关 ln ␬⫺(1⫺1/␬ ) 兴 .
They did not consider a growing inclusion and had, therefore, no
plastic work in the region 0⭐r⭐R.
We can now compare the total mechanical energy W T produced
during generation of an inclusion with the radius R,
冉冊
2 ␬ ln ␬ ⫹2 ␬ ⫺1
␬2
and W p /W Tc
␴ 2f 4R 3 ␲
•2 共 1⫺ ␯ 兲 ␬ ln ␬ .
W p⫽ •
E
3
W T ⫽U⫹W p ⫽
W T /W Tc ⫽
3
eM ,
(8c)
(8d)
(8e)
are depicted in Fig. 2.
W T /W Tc possesses a maximum of 1.456 at ␬ ⫽1.763 and
reaches the value 1 at ␬ ⫽5.03.
For values of ␬ ⬎5.03 the ratio W T /W Tc tends to be smaller than
1, going to 0 as ␬ goes to infinity. The conclusion is that depending on ␬ an elastic-plastic body may consume more or less deformation energy compared with a purely elastic body.
Earmme et al. 关12兴 offered also a solution for the stress and
strain field around a spherical inclusion with fixed radius in a
hardening material. Carrol 关13兴 investigated independently some
further types of hardening laws. In particular, for linear strain
hardening it is interesting to note that the size of the plastic zone
remains unaffected by the hardening. Taking into account the
hardening factor f ⫽1⫹2(1⫺ ␯ )E p /E and denoting the solution
for no hardening, f ⫽1, 共see 共3a–g兲兲 as ␴n , ␧e,n , etc., the following relations can be derived at the interface:
• The inner elastic zone at r⫽R(t)
1
2␴ f
␴⫽ ␴n ⫺
(10a)
•2E p /E• 共 1⫺ ␯ 兲 ␬ e,
f
3f
E
2␴ f
E␧e ⫽ ␧e,n ⫺
•2E p /E• 共 1⫺2 ␯ 兲共 1⫺ ␯ 兲 ␬ e, (10b)
f
3f
␴2f
1
•E /E•共1⫺2␯兲
U⬘⫽ 2 Un⬘⫹8
f
3E p
再
冎
⫻共1⫺␯兲•关1⫹ln ␬⫹Ep /E共1⫺␯兲␬兴 .
(10c)
DECEMBER 2000, Vol. 67 Õ 795
• The elastic-plastic zone at r⫽R(t)
1
2␴ f
␴⫽ ␴n ⫺
(10d)
•2E p /E• 共 1⫺ ␯ 兲 ␬ eM ,
f
3f
E
2␴ f
E␧e ⫽ ␧e,n ⫺
•E p /E• 共 1⫺ ␯ 兲共 1⫹ ␯ 兲 ␬ eM , (10e)
f
3f
␴2f
1
U⬘⫽ 2 U⬘n⫹4
•E /E•共1⫹␯兲共1⫺␯兲•关␬⫹Ep /E•共1⫺␯兲␬2兴 .
f
3E p
(10f)
再
冎
2.2 Thermodynamics of Interface Motion. If one wants to
calculate the thermodynamic force on the interface r⫽R(t), its
mechanical contribution F M can be calculated as the normal component of the Eshelby energy momentum tensor, see, e.g., Fischer
and Reisner 关14兴 Eq. 共26兲. With 冀.冁 denoting the jump of a quantity and 具.典 the average of a quantity at the interface r⫽R(t), one
gets
F M ⫽ 冀 U ⬘ 冁 ⫺ 具 ␴ 典 • 冀 ␧冁 .
T
(11a)
The jump in strain energy 冀 U ⬘ 冁 can be reformulated with the
identity 冀 aT •b冁 ⫽ 具 aT 典 • 冀 b冁 ⫹ 冀 aT 冁 • 具 b典 as
冀 U⬘冁⫽
1 T
1
具 ␴ 典 • 冀 ␧e 冁 ⫹ 冀 ␴T 冁 • 具 ␧e 典 ⫽ 具 ␴T 典 • 冀 ␧e 冁
2
2
(11b)
due to 具 ␴T 典 冀 ␧e 冁 ⫽ 具 ␧Te 典 冀 ␴冁 .
冀␧冁 is 冀 ␧e 冁 ⫺␧ o e, since no jump in the plastic strain ␧p occurs at
the interface. Inserting 冀␧冁 and 共11b兲 into 共11a兲 yields
F M ⫽␧ o 具 ␴T 典 •e.
(11c)
If one introduces an additional load stress ␴ o , F M finally yields
F M ⫽3 ␴ o ␧ o ⫺␧ o
␴f
关共 1⫹2 ln ␬ 兲 ⫹2E p /E• 共 1⫺ ␯ 兲 ␬ 兴 .
f
(11d)
The contribution F M ⫽3 ␴ o ␧ o may be positive or negative depending on the sign of ␴ o . The second part of F M can be considered as a barrier 共due to its sign兲,
F B,int⫽
␴ 2f 共 1⫺ ␯ 兲 ␬
•
关共 1⫹2 ln ␬ 兲 ⫹2E p /E• 共 1⫺ ␯ 兲 ␬ 兴 ,
E
f
(11e)
due to the generation of the strain and stress by the accommodation of the transformation volume strain ␧ o e. The quantity F B,int is
depicted in Fig. 2. Only a rather weak influence of E p /E can be
observed.
The total mechanical driving force F D follows by adding the
jump of the chemical free energy 冀 ␳␸ ch 冁 to F M . As a further
barrier the energy necessary to rebuild the lattice, F L , as well as
to form a new surface, ␥ /R, ␥ specific surface energy, must be
considered for the energy balance at the interface.
Finally, a driving force F D
F D ⫽ 冀 ␳␸ ch 冁 ⫹3 ␴ o ␧ o
(11f)
and a barrier F B with 共11e兲
F B ⫽F B,int⫹F L ⫹ ␥ /R
(11g)
can be derived.
The interface velocity ␻ can then be formulated as a function
796 Õ Vol. 67, DECEMBER 2000
␻ ⫽ ␾ 共 F D /F B 兲 for F D /F B ⭓1, else 0
(11h)
see, e.g., Rosakis and Knowles 关15兴.
2.3 Remark. Recently Durban and Fleck 关16兴 presented a
solution for an expanding spherical cavity in a Drucker-Prager
material with the yield condition
␴ ␪⫺ ␴ r⫹
␮ T
␴ •e⫽ ␴ f ,
3
(12)
where ␮ is a pressure sensitivity material parameter.
Surprisingly their solution for R p , see 共2兲, and ␴ cannot be
reduced to the solution 共3a兲–共3g兲 for ␮ ⫽0. Their ␴ in the plastic
zone leads to a constant stress vector for ␮ ⫽0 which contradicts
the variation of ␴ with r given in relation 共3c兲.
3
Closing Comments
A full set of analytical expressions is given for the deformation,
strain and stress state due to the dilatational growth of a spherical
inclusion in an elastic-plastic material. The solutions can also be
applied for a spherical cavity with the pressure p̃⫽2 ␴ f /3•(1
⫹ln ␬) at the cavity surface. In addition the dissipated plastic
work as well as the thermodynamic force on the transformation
front are presented. The set of equations allows further metalphysical considerations with respect to the nucleation and growth
of precipitations in an elastic-plastic material.
References
关1兴 Roitburd, A. L., and Temkin, D. E., 1986, ‘‘Plastic Deformation and Thermodynamic Hysteresis in Phase Transformation in Solids,’’ Sov. Phys. Solid
State, 28, pp. 432–436.
关2兴 Roitburd, A. L., and Temkin, D. E., 1986, ‘‘Hysteresis of a Phase Transformation in an Elastoplastic Medium,’’ Sov. Phys. Dokl., 31, pp. 414–416.
关3兴 Kaganova, I. M., and Roitburd, A. L., 1989, ‘‘Effect of Plastic Deformation on
the Equilibrium Shape of a New-Phase Inclusion and Thermodynamic Hysteresis,’’ Sov. Phys. Solid State, 31, pp. 545–550.
关4兴 Nazyrov, I. R., and Freidin, A. B., 1998, ‘‘Phase Transformations in Deformation of Solids in a Model Problem of an Elastic Ball,’’ Mech. Solids, 33, pp.
39–56 Izevestiya AN.MTT, 5, pp. 52–71.
关5兴 Vandermeer, R. A., 1990, ‘‘Modeling Diffusional Growth During Austenite
Decomposition to Ferrite in Polycrystalline Fe-C Alloys,’’ Acta Metall.
Mater., 38, pp. 2461–2470.
关6兴 Lee, J. K., Earmme, Y. Y., Aronson, H. J., and Russel, K. C., 1980, ‘‘Plastic
Relaxation of the Transformation Strain Energy of a Misfitting Spherical Precipitate: Ideal Plastic Behavior,’’ Metall. Trans. A, 11A, pp. 1837–1847.
关7兴 Fischer, F. D., Berveiller, M., Tanaka, K., and Oberaigner, E. R., 1994, ‘‘Continuum Mechanical Aspects of Phase Transformations in Solids,’’ Arch. Appl.
Mech., 64, pp. 54–85.
关8兴 Lubliner, J., 1990, Plasticity Theory, Macmillan, New York.
关9兴 Budiansky, B., 1959, ‘‘A Reassessment of Deformation Theories in Plasticity,’’ ASME J. Appl. Mech., 81, pp. 259–264.
关10兴 Franciosi, P., Lormand, G., and Fougeres, R., 1998, ‘‘On Elastic Modeling of
Inclusion Induced Microplasticity in Metallic Matrices From the Dilatating
Sphere Problem,’’ Int. J. Plast., 14, pp. 1013–1032.
关11兴 Böhm, H. J., Fischer, F. D., and Reisner, G., 1997, ‘‘Evaluation of Elastic
Strain Energy of Spheroidal Inclusions With Uniform Volumetric and Shear
Eigenstrains,’’ Scr. Mater., 36, pp. 1053–1059.
关12兴 Earmme, Y. Y., Johnson, W. C., and Lee, J. K., 1981, ‘‘Plastic Relaxation of
the Transformation Strain Energy of a Misfitting Spheroidal Precipitate: Linear
and Power-Law Strain Hardening,’’ Metall. Trans. A, 12A, pp. 1521–1530.
关13兴 Carrol, M. M., 1985, ‘‘Radial Expansion of Hollow Spheres of Elastic-Plastic
Hardening Material,’’ Int. J. Solids Struct., 21, pp. 645–670.
关14兴 Fischer, F. D., and Reisner, G., 1998, ‘‘A Criterion for the Martensitic Transformation of a Microregion in an Elastic-Plastic Material,’’ Acta Mater., 46,
pp. 2095–2102.
关15兴 Rosakis, Ph., and Knowles, J. K., 1997, ‘‘Unstable Kinetic Relations and the
Dynamics of Solid-Solid Phase Transitions,’’ J. Mech. Phys. Solids, 45, pp.
2055–2081.
关16兴 Durban, D., and Fleck, N. A., 1997, ‘‘Spherical Cavity Expansion in a
Drucker-Prager Solid,’’ ASME J. Appl. Mech., 64, pp. 743–750.
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