Download A Multiscale Finite-element Method for Modeling Fully

A Multiscale Finite-element Method for Modeling
Fully-Coupled Thermomechanical Problems in Solids
Arkaprabha SENGUPTA∗, Panayiotis PAPADOPOULOS†and Robert L. TAYLOR‡
University of California, Berkeley, CA 94720, USA
Table of contents
1 Introduction
2
2 Balance laws and constitutive assumptions
3
3 Macroscale and microscale problems
4
4 Boundary conditions for RVE
5
5 Average referential thermomechanical quantities
in the two-scale problem
5.1 The weak forms for the macroscale . . . . . . . . .
5.2 The purely mechanical homogenization problem . .
5.3 Fully-coupled thermomechanical homogenization .
and coupled tangents
. . . . . . . . . . . . . .
. . . . . . . . . . . . . .
. . . . . . . . . . . . . .
7
7
8
10
6 Average spatial thermomechanical quantities and coupled tangents in the
14
two-scale problem
7 Algorithmic implementation
7.1 A unified computational representation of the macroscopic tangent moduli .
7.2 A simple scheme for parallel processing . . . . . . . . . . . . . . . . . . . . .
16
16
17
8 An application to Nitinol polycrystal modeling
8.1 A thermomechanical model for phase transformations at the microscale . .
8.2 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.1 Textured tube under tension . . . . . . . . . . . . . . . . . . . . . .
18
18
20
20
9 Conclusions
22
Bibliography
23
Appendix: Expressions for the macroscopic referential thermomechanical
tangent moduli
24
Abstract
This article proposes a two-scale formulation of fully-coupled continuum thermomechanics using the finite element method at both scales. A monolithic approach is
adopted in the solution of the momentum and energy equations. An efficient implementation of the resulting algorithm is derived that is suitable for multi-core processing.
The proposed method is applied with success to a strongly coupled problem involving
shape-memory alloys.
Keywords: thermomechanics; multiscale; monolithic; finite element; shape-memory alloys
∗
Department of Mechanical Engineering
Department of Mechanical Engineering, corresponding author
‡
Department of Civil and Environmental Engineering
†
1
Multiscale model for thermomechanical response
1
Introduction
In recent years, with the availability of affordable and increasingly powerful multi-processor
computational resources, it has become possible to perform finite element modeling of
materials at two different continuum scales and use fine-scale results to predict macroscopic
state variables and material properties. This may be accomplished by first simulating the
microscale behavior on an appropriately chosen Representative Volume Element (RVE)
comprised of the various microstructural heterogeneities. The local macro-constitution
is subsequently deduced from the microscopic response by an averaging procedure, see,
e.g., [1–5] for early developments. Such FE2 multiscale methods have been employed with
success in modeling complex systems such as composites and polycrystalline materials
undergoing plastic deformation, thermomechanical shock loading or phase transition, see,
e.g., [6–9].
Multiscale thermomechanical problems that have been pursued in the literature vary
not only in application, but also in the precise nature of the coupling between the momentum and energy equations at the two scales. Certain works assume one-way coupling,
where the momentum equations are dependent on the energy equation through temperature. In this case, the momentum equations may be solved via a non-local asymptotic
homogenization approach, as done in certain problems of poro-thermoelasticity [10] and
thermal shock waves [11]. Other works focus on solving the thermomechanical equations
in an uncoupled (staggered) manner using first-order homogenization techniques and the
FE2 method, see [8]. The full two-way thermomechanical coupling, which is physically
present in many dissipative processes under dynamic loading, including those involved in
the plastic deformation of metals and in martensitic phase transformations, has not been
addressed to date within the context of the FE2 method.
This work introduces a novel theoretical and algorithmic treatment of of the fully
coupled multiscale thermomechanical problem using first-order homogenization. On the
theoretical front, stress and heat flux are determined from the microscale using an averaging
procedure, as in earlier works. However, in order to account for the effect of dissipation
on the macroscale energy equation, an additional cumulative heat supply term is defined
in the microscale and its macroscopic counterpart is determined by averaging over the
RVE. This enables the accurate conveyance of the thermomechanical coupling effects from
the microscale to the macroscale. On the computational front, a monolithic approach is
adopted in both scales to account for strong coupling between the momentum and energy
2
Version: September 6, 2011, 14:03
A. Sengupta, P. Papadopoulos and R.L. Taylor
equations. The algorithmic implementation extends earlier work on purely mechanical
homogenization [4] in computing all the coupled macroscopic tangent moduli from the
converged microscopic stiffness matrix using a computationally efficient static condensation
procedure. Given the computational complexity of FE2 algorithms, attention is also focused
on a simple parallel implementation that is well-suited for systems of multi-core processors.
An application to a phase-transforming polycrystal under dynamic loading is included to
demonstrate the applicability of the proposed FE2 method.
The organization of the paper is as follows: The governing equations of continuum thermomechanics are summarized in Section 2. This is followed in Section 3 by an exposition of
the principal assumptions relevant to the macroscale and microscale. Section 4 reviews the
boundary conditions for the thermomechanical RVE. A detailed derivation of homogenized
referential and spatial thermomechanical quantities is included respectively in Sections 5
and 6. Next, the algorithmic implementation is presented in Section 7, followed by the
application to Nitinol polycrystal modeling in Section 8. Concluding remarks are finally
offered in Section 9.
2
Balance laws and constitutive assumptions
The linear and angular momentum balance equations are written in local referential form
as
ρ0 a = DivP + ρ0 b ,
(1)
PFT = FPT ,
where P is the first Piola-Kirchhoff stress tensor, ρ0 is the mass density per unit referential
volume, b is the body force per unit mass, a is the acceleration, and Div denotes divergence
relative to the referential coordinates. In addition, the energy balance equation takes the
form
ρ0 ˙ = ρ0 r − Divq0 + P · Ḟ ,
(2)
in terms of the internal energy per unit mass , the heat supply per unit mass r, and the
heat flux vector q0 resolved over the geometry of the reference configuration. The above
equation may be written in concise form as
cθ̇ = − Div q0 + r̂ ,
(3)
∂
and r̂ is comprised of the external heat supply ρ0 r, the stress power
∂θ
P · Ḟ, and any contributions due to the dependence of the internal energy on quantities
where c = ρ0
3
Version: September 6, 2011, 14:03
Multiscale model for thermomechanical response
other than temperature. The multiscale framework for thermomechanical problems will be
developed based on the energy equation in (3).
Finally, a local referential form of the Clausius-Duhem inequality may be stated as
ρ0 η̇θ ≥ ρ0 r − Div q0 +
q0 · Grad θ
,
θ
(4)
where η is the entropy per unit mass and “Grad” denotes gradient relative to the referential
coordinates.
In this work, it is assumed from the outset that the stress tensor P and the cumulative
heat supply r̂ may depend only on the deformation and the temperature (including their
histories), while the heat flux vector q0 depends only on the temperature gradient.
3
Macroscale and microscale problems
Solving the fully-coupled thermomechanical two-scale IBVP involves satisfying the linear
momentum balance equation (1)1 and the energy equation (3) at both scales, and then
imposing the necessary conditions to effect the handshake between the two scales. In this
section, suitable forms of the momentum and energy equations are introduced for each
scale. To distinguish between the two scales, superscripts “m” and “M ” will be used to
denote physical quantities at the microscale and macroscale, respectively.
The local referential forms of the macroscale balances of linear momentum and energy
are stated as
M
M
ρM
= Div PM + ρM
0 b
0 a
,
M
cM θ̇ M = − Div qM
.
0 + r̂
(5)
Here, r̂ M is the cumulative macroscopic heat supply per unit volume due to external heat
source, mechanical dissipation and thermoelastic coupling.
It is assumed next that the characteristic time and length in the microscale are much
smaller than their counterparts in the macroscale. Specifically, the time needed to impose
changes to the mechanical and thermal loading at the macroscale is assumed to be much
larger than the time required for the velocity and temperature of the microscopic problem
to reach steady-state due to such changes. As a result, one may ignore the acceleration
term in the momentum equation and the term involving the rate of temperature change
in the energy equation. Note that a steady-state at the microscale refers here to the
time-independent thermomechanical state reached by the microscale after an incremental
change to the macroscopic state [8]. This is an instantaneous “steady-state” defined for
4
Version: September 6, 2011, 14:03
A. Sengupta, P. Papadopoulos and R.L. Taylor
a given macrostate, although the latter is itself time-varying. Likewise, the size of the
macroscopic specimen is taken to be much larger than the size of the region to be modeled
microscopically. This implies that the body force in the momentum equation is negligible
compared to the stress-divergence term the microscale, see [12]. In addition, the cumulative
heat supply (which includes here the stress power) may be neglected compared to the fluxdivergence term in the microscale as long as there is length-scale separation and the rate of
macroscopic loading remains bounded. Under the preceding circumstances, the divergence
terms in (1)1 and (3) become dominant in the microscale, hence the balance equations
corresponding to (5) simplify to
Div Pm = 0
− Div qm
0 = 0 .
,
(6)
Note that, while the body force and heat generation are negligible when determining the
instantaneous microscale response, their effect may be significant in the macroscopic problem. Hence, these quantities are accounted for at the macroscale through homogenization,
as will be seen in Section 5.3.
In the two-scale problem discussed here, FM , θ M and Grad θ M are passed from the
M are determined by
M
M
M
macroscale to the microscale, while PM , qM
0 , r̂ , b , ρ0 and c
homogenization from the respective microscale quantities and returned to the macroscale.
4
Boundary conditions for RVE
This section explores the relation between microscopic and macroscopic quantities, as formulated through the choice of boundary conditions in the microscale. Average values over
¯
the representative volume element (RVE) of the microscale are denoted by (·).
It is easy to show with the aid of (6)1 and the divergence theorem that
Z
Z
1
1
Pm · Fm dV − P̄ · F̄ =
(pm − P̄Nm ) · (xm − F̄Xm ) dA ,
V ω0
V ∂ω0
(7)
where ω0 is the reference domain of the RVE with volume V , ∂ω0 is the boundary of
ω0 with outward unit normal Nm , Xm , xm are the referential and spatial positions of
a material in the microscale, and pm = Pm Nm is the Piola traction. If the macroscopic
deformation gradient and first Piola-Kirchhoff stress are taken to be equal to the average of
the respective microscopic quantities (namely, if FM = F̄ and PM = P̄), then the relation
Z
(pm − PM Nm ) · (xm − FM Xm ) dA = 0
(8)
∂ω0
5
Version: September 6, 2011, 14:03
Multiscale model for thermomechanical response
implies that
1
V
Z
Pm · Fm dV = PM · FM .
(9)
ω0
Equation (9) corresponds to the classical Hill-Mandel macro-homogeneity condition [13].
The relation (8) may be enforced by means of microscale boundary displacement or
traction boundary conditions, such that
xm − FM Xm = 0
(10)
pm − PM Nm = 0 .
(11)
or
Equation (8) is also satisfied by periodic boundary conditions defined as
x+ = x− + FM (X+ − X− ) ,
p+ = −p− ,
(12)
where the superscripts “+” and “−” signify points which lie on opposite faces of a periodic
RVE. These conditions enforce (8) due to the fact that the normals at two corresponding
points on opposite faces are taken to be equal and opposite of each other, i.e., Nm,+ =
−Nm,− . Figure 1 illustrates the deformation of an RVE under such periodic boundary
conditions.
A displacement-like thermal boundary condition may be imposed on the boundary of
the RVE as
θ = θ M + Grad θ M · (Xm − Xc ) .
(13)
Here, Xc denotes the position vector of the centroid of the RVE in the reference configuration. The preceding boundary condition ensures that the temperature at the boundary
of the RVE varies linearly according to the macroscopic temperature gradient Grad θ M . In
addition, this linear boundary field, if extended linearly to ω0 , would maintain an average
temperature equal to its value at Xc , which is precisely the macroscopic temperature θ M .
Therefore, given that the energy balance within the RVE involves no heat supply (see (6)2 ),
condition (13) plays the role of a “thermostat” that keeps the average RVE temperature
close to θ M without strongly enforcing this condition, as done in [14]. In fact, the latter
condition is recovered naturally from (13) when the heat transport properties in the microscale are uniform. Alternatively, periodic boundary conditions may be imposed, where
the boundary nodes are linked through the conditions
θ + = θ − + Grad θ M · (Xm − X− )
6
,
m
m
q+
= −q−
0 ·N
0 ·N ,
(14)
Version: September 6, 2011, 14:03
A. Sengupta, P. Papadopoulos and R.L. Taylor
in analogy to (12). As in the preceding mechanical equations, it is easy to show that if
qM
0 is chosen to be equal to the average of the microscale flux over the domain of the RVE
(i.e., if qM
0 = q̄0 ), then a Hill-Mandel-like condition for the dot-product q0 · Grad θ would
be enforced by either (13) or (14), as shown in [14, 15]. However, such a justification of
these boundary conditions is not advocated here, because the term q0 · Grad θ does not
fully describe the internal entropy production for general thermomechanical processes, in
contrast to the mechanical problem where the term P · Ḟ fully describes the stress power.
5
Average referential thermomechanical quantities and coupled tangents in the two-scale problem
In this paper, a monolithic approach is advocated for the solution of the coupled thermomechanical equations in both scales. To this end, Section 5.1 includes a summary of
the weak forms for the governing equations in the macroscale. Next, Section 5.2 uses
the finite element equations to deduce expressions for stress and tangent moduli in the
macroscale by temporarily suppressing the thermal effects in the microscale. The latter
effects are subsequently incorporated in Section 5.3, where the average thermal conductivities and cumulative heat supply are also calculated to complete the description of the
coupled two-scale thermomechanical problem.
5.1
The weak forms for the macroscale
Let Ω0 be the region occupied by the body in the reference configuration of the macroscale,
and denote by ∂Ω0 its smooth and orientable boundary with outward unit normal NM .
Assume that ∂Ω0 be decomposed into parts ΓuD,0 and ΓuN,0 , where Dirichlet and Neumann
boundary conditions are enforced for the momentum equations (5)1 , as well as into parts
ΓtD,0 and ΓtN,0 , where Dirichlet and Neumann boundary conditions are enforced for the
energy equation (5)2 . For the preceding decompositions, the conditions ΓuD,0 ∪ ΓuN,0 =
ΓtD,0 ∪ ΓtN,0 = ∂Ω0 are assumed to hold.
The weak form of linear momentum balance can be expressed as
Z
Z
Z
∂w
M
M
M
w · ρM
− aM ) dV ,
w · p̄ dA +
· P dV =
0 (b
M
u
∂X
Ω0
Ω0
ΓN,0
(15)
where w is a H 1 -smooth vector weighting function that vanishes on ΓuD,0 and p̄M = PM NM
is the prescribed traction resolved on the geometry of the reference configuration. The weak
7
Version: September 6, 2011, 14:03
Multiscale model for thermomechanical response
form of the energy equation is likewise derived as
Z
Ω0
wcM θ̇ M dV +
Z
Ω0
Grad w · qM
0 dV
=
Z
wr̂ M dV −
Ω0
Z
wh̄M dA , (16)
ΓtN,0
M
where w is a H 1 -smooth scalar weighting function that vanishes on ΓtD,0 and h̄M = qM
0 ·N
is the prescribed heat flux on ΓtN,0 .
5.2
The purely mechanical homogenization problem
In the interest of clarity, the procedure for deducing macroscopic stress and tangent moduli
from the microscale finite element problem is first described for the purely mechanical
problem following the procedure originally derived in [4]. This preliminary step will greatly
facilitate the presentation of the fully-coupled thermomechanical case in Section 5.3. Note
that indicial and matrix notation are used here interchangeably, as deemed necessary, to
elucidate the derivations.
The displacement and periodic mechanical boundary conditions discussed in Section 4
may be applied to the boundary nodes of an RVE mesh to deduce an expression for the
average first Piola-Kirchhoff stress in terms of a sum over nodal forces. Specifically, with
the aid of (6)1 , the relation p = Pm Nm , and the divergence theorem, it follows that
Z
Z
1
1 X E E
1
m
M
XB fa ,
(17)
PaB dV =
XB pa dA =
PaB =
V ω0
V ∂ω0
V
E∈ext
where “ext” stands for the set of boundary nodes of the RVE. In equation (17), faE are
components of the equivalent nodal force at node E of the RVE boundary obtained from
E are
the nodal reactions required to satisfy equilibrium in the microscale. Likewise, XB
components of the position vector of the boundary node E in the reference configuration.
Next, let the displacement degrees-of-freedom (DOFs) be partitioned into internal and
I
external, denoted in indicial form by uIa and uE
a and in aggregated matrix form by [u ]
and [uE ], respectively. It follows that, when equilibrium is satisfied, the residual forces
associated with the internal DOFs vanish, hence
[C JI ][δuI ] + [C JE ][δuE ] = [0] ,
I, J ∈ int , E ∈ ext ,
(18)
where [δuI ] and [δuE ] denote variations of the internal and external displacements and
[C JI ], [C JE ] are submatrices of the tangent stiffness matrix [C] associated with the internalinternal and internal-external DOFs, respectively. In addition, “int” in (18) signifies the
8
Version: September 6, 2011, 14:03
A. Sengupta, P. Papadopoulos and R.L. Taylor
set of internal nodes in the RVE. It follows that if there are Nint internal nodes and Next
external nodes in the RVE, the dimensions of [δuI ] and [δuE ] are 3*Nint and 3*Next,
respectively. Assuming now that the mechanical problem is well-posed, the variations of
the internal DOFs may be extracted from (18) as
[δuI ] = −[C JI ]−1 [C JE ][δuE ] .
(19)
Consistent tangent moduli for the average first Piola-Kirchhoff stress may be obtained
for the case of the displacement boundary conditions of Section 4 by noting that variations
of the macroscale deformation gradient affect the displacement DOFs in the RVE boundary
through (10). Therefore, the variation in displacement components at a typical boundary
node F is given by
M
F
δuFa = δFaB
XB
,
F ∈ ext .
(20)
Further, the variation in the mechanical residual of a boundary node E is given in component form by
EF
EI
EF
δuFb ,
δuIb = Ĉab
δuFb + Cab
δfaE = Cab
E, F ∈ ext , I ∈ int ,
(21)
EF are the components of the 3×3 submatrix of the tangent stiffness resulting
where Ĉab
from the static condensation of the internal DOFs, and defined with the aid of (19) as
[Ĉ EF ] = [C EF ] − [C EI ][C JI ]−1 [C JF ] ,
E, F ∈ ext , I, J ∈ int .
(22)
Now, combining equations (17)3 , (20) and (21), the macroscopic mechanical tangent modulus may be written in component form as
ĀaBbC =
M
∂PaB
1 X X E EF F
XB Ĉab XC .
=
M
V
∂FbC
E∈ext F ∈ext
(23)
see also [4].
A computationally efficient scheme for computing the preceding macroscopic tangent
modulus involves first defining the matrices [GI ], [G̃J ] and [H] with components
GIaBb =
X
E EI
XB
Cab
E∈ext
G̃JabC
=
X
JF F
XC
Cab
(24)
F ∈ext
HaBbC =
X X
E EF F
XB
Cab XC ,
E, F ∈ ext , I, J ∈ int ,
E∈ext F ∈ext
9
Version: September 6, 2011, 14:03
Multiscale model for thermomechanical response
and then noting that (23) can be expressed as
X X
1
[H] −
[GI ][C JI ]−1 [G̃J ] .
[Ā] =
V
(25)
I∈int J∈int
Equations (24) and (25) imply that, in general, [Ā] is not symmetric.
In the case of periodic boundary conditions, the corner nodes are deformed according to
the macroscopic deformation gradient, as in (10), while the remaining boundary nodes are
linked by the periodic condition (12)1 . Since equation (12)2 stipulates that the tractions
are antiperiodic, equation (17) yields the macroscale Piola-Kirchhoff stress as


X
X
1 
E− E 
M
E E
E
PaB
=
)fa
XB
fa +
(XB
− XB
,
V
(26)
E∈p+
E∈fix
where “fix” and “p+” denote, respectively, the set of corner (fixed) nodes and nodes on
E to be equal to
the “+” face linked through the periodicity conditions (12). Defining X̂B
E − X E− ) for the “p+” nodes and to X E for the “fix” nodes, equation (26) may be
(XB
B
B
rewritten compactly as
M
PaB
=
1
V
X
E E
X̂B
fa ,
(27)
E∈fix,p+
where “fix,p+” denotes the union of all nodes in the “fix” and “p+” sets. Further, recalling
(12)1 , the variation of boundary DOFs is given by
M F
X̂B ,
δuFb = δFbB
F ∈ fix,p+ .
(28)
In analogy to (23), the mechanical tangent moduli for the periodic case is obtained in
component form as
ĀaBbC =
M
∂PaB
1
=
M
V
∂FbC
X
X
E EF F
X̂B
Ĉab X̂C .
(29)
E∈fix,p+ F ∈fix,p+
EF are components of the 3×3 submatrix of the tangent stiffness obtained using
Here, Ĉab
(22), where now static condensation is imposed on all internal nodes, as well as on the
“p-” external nodes. Again, the macroscopic tangent modulus may be expressed as in 25,
E used in (24) are now replaced by their
the only difference being that the coordinates XA
E.
periodic counterparts X̂A
5.3
Fully-coupled thermomechanical homogenization
In this section, the full coupling between the mechanical and thermal problems is delineated
and a procedure is proposed for determining the macroscopic flux, cumulative heat supply,
and heat capacity, as well as the corresponding tangent moduli.
10
Version: September 6, 2011, 14:03
A. Sengupta, P. Papadopoulos and R.L. Taylor
In the fully-coupled thermomechanical case, the expanded DOFs include both displacements and temperature, jointly expressed in the vector d = (u, θ). It is important to note
that variations to the interior nodal DOFs are related to their exterior counterparts according to
[δdI ] = −[C JI ]−1 [C JE ][δdE ] ,
(30)
of notation
in complete analogy to (19) and with a slight abuse
 in retaining the nomen
[Cuu ] [Cuθ ]
 for the extended set of
clature for the tangent stiffness matrix [C] = 
[Cθu ] [Cθθ ]
DOFs.
The variation of the mechanical residual for the external nodes is suitably extended to
the thermomechanical problem from (21) as
EF
EF
δfaE = Ĉuu;ab
δuFb + Ĉuθ;a
δθ F ,
E, F ∈ ext ,
(31)
where the condensed stiffness [Ĉ EF ] is also extended from its purely mechanical counterpart
in (22). Also, subscripts ‘uu’ and ‘uθ’ are introduced in equation (31) to distinguish
displacement-displacement from displacement-temperature stiffness terms, the latter being
due to thermal coupling in the momentum equations. In addition, a semicolon is used to
separate the u- and/or θ-subscripts from the indices signifying displacement components.
For the case of the displacement-like temperature boundary conditions in (13), the variations in boundary temperatures can be expressed in terms of the independent variations
of θ M and Grad θ M as
F
c
δθ F = δθ M + δ(Grad θ)M
A (XA − XA ) ,
F ∈ ext .
(32)
Therefore, using (17)3 , (31) and (32), one finds the macroscopic tangent moduli due to
thermal expansion to be
M
∂PaB
1 X X E EF
XB Ĉuθ;a .
=
∂θ M
V
(33)
E∈ext F ∈ext
Following the discussion in Section 4, the macroscopic heat flux is determined as a
volume average over the RVE. Indeed, recalling (6)2 , one may write
Z
Z
1
1
1 X E E
M
m
q0 A =
XA q0 n ,
q0 A dV =
XA (q0mB NBm ) dA =
V ω0
V ∂ω0
V
(34)
E∈ext
where q0En is the equivalent normal heat flux on the boundary nodes obtained from the
boundary residual of the energy equation. Note that the macroscopic heat flux is obtained here by applying a constant temperature gradient on the RVE boundary and then
11
Version: September 6, 2011, 14:03
Multiscale model for thermomechanical response
evaluating the resulting normal fluxes on the same boundary. This lends physical justification to adopting (34)1 as a principal homogenization assumption in the proposed multiscale approach as opposed to admitting a Hill-Mandel-like volume-averaging condition
on q0 · Grad θ, which only partly represents the internal entropy production in a general
thermomechanical process.
Taking into account (34)3 , the variation of the boundary residual for the energy equation
can be written as
EF
EF
EF
δθ F .
δθ F = Ĉθθ
δuFa + Ĉθθ
δq0En = Ĉθu;a
(35)
EF in (35) vanishes because, by assumption, the energy equation
Note that the term Ĉθu;a
(6)2 in the microscale does not depend explicitly on deformation. Therefore, appealing to
(34), (35) and (32), the referential macro-conductivity is written in component form as
M
KAB
=
∂q0MA
1 X X E EF F
c
XA Ĉθθ (XB − XB
).
=
M
∂(Grad θ )B
V
(36)
E∈ext F ∈ext
As argued in Section 3, the macroscopic cumulative heat supply r̂ M , body force bM ,
M are taken to be equal to the volume averages of the
mass density ρM
0 and heat capacity c
respective microscopic quantities, namely,
Z
Z
1
1
m
M
M
r̂ dV , b =
bm dV ,
r̂
=
V ω0
V ω0
Z
1
ρm dV
ρM
=
0
V ω0 0
,
cM =
1
V
Z
cm dV . (37)
ω0
In particular, (37)4 is consistent with the findings in [16], where an asymptotic expansion
method is applied to estimate effective thermal properties of periodic composites. Also,
note that, for implementational purposes, the rate terms in the microscale variable r̂ m of
(37)1 are computed using a first-order finite difference approximation of the participating
time derivatives in an average sense over the macroscale time-step, see Section 8.1 for
specific examples of such rate terms.
The tangent moduli for the macroscale energy equation can be determined by first
noting that the variations of the microscale deformation gradient are expressed in terms of
variations of nodal displacements as
m
δFaB
=
X
E
φE
,B δua +
E∈ext
X
φI,B δuIa ,
(38)
I∈int
where φI and φE are interpolation functions corresponding to internal and external nodes,
respectively. In addition, the variation of the microscale temperature can be similarly
12
Version: September 6, 2011, 14:03
A. Sengupta, P. Papadopoulos and R.L. Taylor
expressed as
δθ m =
X
φE δθ E +
E∈ext
X
φI δθ I .
(39)
I∈int
Now, the variation of r̂ M in terms of the variations of the microscale nodal displacements
and temperatures is obtained with the aid of (37)1 , (38) and (39) as
δr̂
M
Z
Z
∂r̂ m E
∂r̂ m E
1 X
E
E
φ dV δθ
δua +
=
m φ,A dV
m
V
ω0 ∂FaA
ω0 ∂θ
E∈ext
Z
Z
1 X
∂r̂ m I
∂r̂ m I
I
I
+
φ dV δθ
δua +
. (40)
m φ,A dV
m
V
ω0 ∂FaA
ω0 ∂θ
I∈int
Thus, in the case of displacement-like boundary conditions the tangent moduli for r̂ M are
given by
Z
∂r̂ m E
1 X
∂r̂ M
=
m φ,A dV +
M
V
∂FaA
∂FaB
ω
0
E∈ext
#
X
X Z ∂r̂ m
E
I
−(C JI )−1 C JE uu;ba XB
, (41)
m φ,A dV
ω0 ∂FbA
J∈int
I∈int
and
∂r̂ M
1 X
=
∂θ M
V
E∈ext
"Z
ω0
X
X Z ∂r̂ m
∂r̂ m E
JI −1 JE
I
−(C
)
C
φ
dV
+
φ
dV
θθ
m
∂θ m
ω0 ∂θ
J∈int
I∈int
#
X
X Z ∂r̂ m
I
+
−(C JI )−1 C JE uθ;b . (42)
φ,A dV
m
ω0 ∂FbA
I∈int
J∈int
In the above, the variations in (20) and (32), and appropriate submatrices of (30) have
been used.
The preceding two-scale method may also be employed in the case of the periodic
boundary conditions (12) and (14) in a manner analogous to the purely mechanical case
of Section 5.2. Here, the temperature is prescribed at the corner nodes as in (13), while
the remaining boundary nodes are linked by the periodic condition (14)1 . Also, akin to
the derivation of macroscopic first Piola-Kirchhoff stress in (27), the referential heat flux
for the periodic case is obtained as
q0MA =
1
V
X
E E
X̂A
q0 n ,
(43)
E∈fix,p+
where q0En is antiperiodic over the RVE boundary. Note that, owing to the periodicity, the
variations of temperatures for the linked nodes (“p+”) in the microscale do not depend on
13
Version: September 6, 2011, 14:03
Multiscale model for thermomechanical response
the variation of the macroscopic temperature. Hence, the tangent moduli due to thermal
expansion are suitably modified from (33) and take the form
M
∂PaB
1
=
M
∂θ
V
X
X
E EF
X̂B
Ĉuθ;a .
(44)
E∈fix,p+ F ∈fix
Likewise, (41) would be modified by letting the external sum range over “fix,p+” and by
E for X E , while in (42) the external sum would range over “fix” only. Lastly,
substituting X̂B
B
the referential macro-conductivity for the periodic case may be readily derived as a slight
departure from (36) in the form
M
KAB
=
6
∂q0MA
∂(Grad θ M )B
=
1
V
X
E∈fix,p+

E
X̂A
X
EF
F
c
Ĉθθ
(XB
− XB
)+
F ∈fix
X
F ∈p+

F− 
EF
F
) .
Ĉθθ
(XB
− XB
(45)
Average spatial thermomechanical quantities and coupled
tangents in the two-scale problem
The homogenized thermomechanical quantities and the corresponding tangent moduli may
be equivalently derived in the spatial form. In fact, the latter results in considerable
computational savings. Indeed, the macroscopic Cauchy (or Kirchhoff) stress, cumulative
heat supply and corresponding spatial tangent moduli require a total of 6+1+49=56 words,
as opposed to 9+1+100=110 words for their referential counterparts derived in Section 5.3.
These spatial quantities are derived henceforth.
Using (17) and (10), the components of the Kirchhoff stress tensor τ M = PM FM
the displacement-like boundary conditions are given by
Z
Z
1
1
1 X E E
M
m M
τab =
xb f a ,
PaB FbB dV =
xb p m
a dA =
V ω0
V ∂ω0
V
T
for
(46)
E∈ext
where xE
b are the components of the current position vector for node E. Now, the components of the Cauchy stress TM =
1
τM
JM
M
Tab
=
are obtained from (46) as
1 X E E
xb f a ,
v
(47)
E∈ext
where J M = det FM and v is the spatial volume of the RVE related to the corresponding
R
referential volume as v = ω0 J m dV m = V J M by virtue of the relation FM = F̄. Similarly,
14
Version: September 6, 2011, 14:03
A. Sengupta, P. Papadopoulos and R.L. Taylor
recalling (34)3 , the macroscopic spatial heat flux qM =
1
FM qM
0
JM
is written in component
form as
1 X E E
xa q 0 n .
v
qaM =
(48)
E∈ext
Hence, the spatial macro-conductivity is obtained for the displacement-like boundary conditions as
∂qaM
1 X X E EF F
xa Ĉθθ (xb − xcb ) ,
=
∂(grad θ M )b
v
M
kab
=
(49)
E∈ext F ∈ext
where
xc
=
F M Xc .
The mechanical tangent moduli are obtained
Z by observing that the stress-divergence
∂w M −1 M
term in (15) can be equivalently expressed as
F
· τ dV . It can be easily
M
Ω0 ∂X
shown using the relation between the Kirchhoff and the first Piola-Kirchhoff stress that
the components c̄abcd of the spatial moduli (namely the symmetric moduli obtained by
linearizing the stress-divergence term written in spatial form) are related to the components
ĀaBcD defined in (29) as
1
M
M
M
M
− (δac τbd
c̄abcd = FbB
ĀaBcD FdD
).
+ δad τbc
2
(50)
The tangent moduli for the Kirchhoff stresses due to temperature are now expressed as
M
∂τab
1 X X E EF
(51)
xb Ĉuθ;a ,
=
∂θ M
V
E∈ext F ∈ext
where (33) and (10) are used. Next, the tangent moduli in (41) are related to their
symmetric spatial counterparts c̄ab according to
1 ∂r̂ M M
∂r̂ M M
c̄ab =
F +
F
,
M bB
M aB
2 ∂FaB
∂FbB
(52)
where the chain rule has been employed. Lastly, note that the tangent moduli in (42)
remain unchanged in the spatial formulation.
For the case of periodic boundary conditions, the components of the macroscopic heat
flux in the current configuration become
qaM =
1
v
X
E
x̂E
a q0 n .
(53)
E∈fix,p+
Now, the components of the Kirchhoff and Cauchy stresses are respectively written as
M
τab
1 X
1 X
E
M
E
M
x̂E
f
,
T
=
x̂E
(54)
=
τab
=
b
a
ab
b fa ,
M
V
J
v
E∈fix,p+
E∈fix,p+
where x̂E equals to (xE −xE− ) for the “p+” nodes and xE for the “fix” nodes. The tangent
moduli can be easily derived using (50), (51) and (52).
15
Version: September 6, 2011, 14:03
Multiscale model for thermomechanical response
7
7.1
Algorithmic implementation
A unified computational representation of the macroscopic tangent
moduli
Following the procedure established for the purely mechanical problem in Section 5.2, an
efficient computation of the macroscopic referential thermomechanical tangent moduli [Ă]
for the displacement-like boundary conditions is employed. This involves the reduction of
the microscopic stiffness matrices by multiplying them by the external referential coordinates as done in (24), followed by static condensation on the reduced matrices to yield the
thermomechanical counterpart of (25), which is repeated here for clarity:
X X
1
[H] −
[GI ][C JI ]−1 [G̃J ] .
[Ă] =
V
(55)
I∈int J∈int
All four matrices that enter (55) are schematically depicted in a unified manner in Figure
2. To within the inverse of the volume V , [Ă] equals the Schur complement of [C JI ] in
the combined matrix of Figure 2. The derivation of the ((4*Nint)×(4*Nint)) submatrix
[C JI ] and the ((4*Nint)×13) submatrix [G̃J ] are entirely straightforward, as they are the
microscopic tangent stiffness and its matrix multiplication with referential coordinates of
external nodes. The (13×(4*Nint)) submatrix [GI ], in addition to straightforward terms
I ] and [r I ] defined as
as above, involves the terms [rF
θ
Z
∂r̂ m
I
[ m ][Grad φI ] dV ,
[rF
] =
∂F
Zω0
∂r̂ m
[ m ][φI ] dV .
[rθI ] =
ω0 ∂θ
(56)
Similarly the (13×13) submatrix [H] includes the terms [D]b , [DA]b , [rF ]b , [rθ ]b and [K]b
defined as
[D]b =
X X
EF
[X E ][Cuu
][X F ]T ,
E∈ext F ∈ext
b
[DA] =
X X
EF
],
[X E ][Cuθ
E∈ext F ∈ext
∂r̂ m
E
[ m ][Grad φ ] dV [X E ] ,
[rF ] =
∂F
ω
0
E∈ext
Z
X
∂r̂ m
[ m ][φE ] dV [X E ] ,
[rθ ]b =
ω0 ∂θ
E∈ext
X X
EF
][(X F − X c )]T .
[K]b =
[X E ][Cθθ
b
X Z
(57)
E∈ext F ∈ext
16
Version: September 6, 2011, 14:03
A. Sengupta, P. Papadopoulos and R.L. Taylor
In the above expressions, the superscript “b” is used to signify that they involve sum∂qM
∂qM
∂PM
0
0
,
,
, and
mation over all boundary nodes. In addition, the terms
∂(Grad θ)M ∂FM ∂θ M
∂r̂
which also appear in [H] vanish identically due to the assumed constitutive
∂(Grad θ)M
M
dependencies of PM , qM
O and r̂ . A further simplification arises owing to the independence
of the microscale heat flux from the deformation in (6)2 , which leads to the vanishing of
the term [Cθu ]. Based on the unified computational representation, the derivation of the
coupled tangent moduli whose expressions were obtained in Section 5.3, is elaborated in
the Appendix.
The preceding representation applies also to the case of periodic boundary conditions,
E are replaced by X̂ E , as described in
as long as the external referential coordinates XA
A
Section 5.2. Likewise, the computation of the macroscopic spatial thermomechanical tanE with their spatial
gent moduli involves replacing the external referential coordinates XA
E
E
counterparts xE
a (or, correspondingly, X̂A with x̂a for periodic boundary conditions) and
adding any extra stress terms as in (50).
7.2
A simple scheme for parallel processing
The above multi-scale algorithm is implemented as a parallel extension to the generalpurpose finite element program FEAP [17, 18] using the Open MPI [19] version of the
classical Message Passing Interface. The FEAP elements are divided into two groups: One
for which the constitutive behavior is specified by standard material models (that may be
used as either microscale or macroscale elements) and a second for which the constitutive
behavior is specified by upscaling the homogenized response of RVEs (that are used as
macroscale elements).
The macroscale problem is solved using conventional 8-node hexahedral finite elements
which have displacement and temperature as nodal DOFs. A boundary-value problem is
defined for each RVE using again an 8-node hexahedral finite element discretization on a
cubic domain subject to either displacement-like or periodic boundary conditions. Nonhomogeneous boundary conditions in the microscale are defined by downscaling FM , θ M
and Grad θ M from each quadrature point of the macroscale to the corresponding RVE
model. The present implementation permits multiple independent RVE models. Partitioning of the microscale workload is effected by assigning a group of macroscale quadrature
points to a processor, see Figure 3 for a simple schematic depiction of the code structure.
Generally, an RVE model is used to solve a boundary-value problem for one or more of
17
Version: September 6, 2011, 14:03
Multiscale model for thermomechanical response
the quadrature points of the macroscale problem. For each solution step, it is necessary
to save and retrieve information that pertains to each quadrature point associated with an
RVE model.
The discrete macroscale and microscale problems are solved here using iterative algorithms based on the Newton-Raphson method. For each iteration of the macroscale
analysis, the current values of FM , θ M and Grad θ M are determined at each quadrature
point. This information is communicated to the RVE-assigned processors by an MPI Send
command. Subsequently, the microscale boundary-value problem is solved and the homogenized stress P̄, flux q̄0 and global tangent moduli are computed. This information is
communicated back to the macroscale processor by an MPI Recv command.
8
An application to Nitinol polycrystal modeling
The proposed multiscale method for the fully coupled thermomechanics problem is applied
here to a model of martensitic phase transformation in Nitinol, where the coupling of the
mechanical and thermal components plays an important role in the overall response of the
material to transient loading. Here, the coupling is due to the temperature-dependence
of the phase transformation and to the latent heat exchanged during the transformation
itself.
8.1
A thermomechanical model for phase transformations at the microscale
A constitutive model for phase transformations under thermomechanical loading of Nitinol
is proposed in [20]. Here, the microscale response is obtained by using this model for Nitinol
single-crystals. According to the model, the total deformation gradient F is multiplicatively
decomposed into
F = Fe Ft ,
(58)
where Fe and Ft are the elastic and transformation deformation gradients. The transformation deformation gradient is expressed in terms of the volume fractions ξα , α = 1, 2, . . . , nv,
of with each of the nv martensitic variants as
Ft = F̂t ({ξα }) = I +
nv
X
ξα Htα ,
(59)
α=1
where the transformation displacement gradient Htα for variant α is written in terms of
the habit-plane displacement bα and the corresponding habit-plane normal mα as Htα =
18
Version: September 6, 2011, 14:03
A. Sengupta, P. Papadopoulos and R.L. Taylor
bα ⊗ mα . Given the elastic deformation gradient in (58), the elastic Green-Lagrange strain
Ee relative to the intermediate configuration induced by Ft is
Ee = Êe (E, {ξα }) =
i
1 h t −T t −1
1 eT e
−1
−T
−I ,
F
F
(F F − I) = Ft EFt +
2
2
(60)
1 T
(F F − I)) is the Lagrangian strain. Taking into account the definition of the
2
second Piola-Kirchhoff stress S (= F−1 P), the energy equation (2) may also be written as
where E(=
ρ0 ˙ = ρ0 r − Divq0 + S · Ė .
(61)
The Helmholtz free energy Ψ is defined in terms of the specific internal energy and
entropy η as
Ψ = ρ0 ( − ηθ) .
(62)
Following [20], the free energy Ψ is taken to be of the form
Ψ = Ψ̂(E, {ξα }, θ) =
X
1 e
E · CEe − Ee · CA(θ − θ0 ) + B(θ − θT )
ξα
2
α
θ
, (63)
+ c θ − θ0 − θ log
θ0
where C is the (isotropic) fourth-order elastic modulus tensor, B(> 0) is the chemical
energy coefficient, c is the volumetric heat capacity, and θT and θ0 are the thermodynamic
equilibrium and reference temperatures, respectively. The Coleman-Noll procedure, when
applied to this model, yields the following constitutive equations [20]:
S =
∂ Ψ̂
∂E
,
ρ0 η = −
∂ Ψ̂
∂θ
,
−
X ∂ Ψ̂
ξ̇α ≥ 0 ,
∂ξα
α
q0 · Grad θ
≤ 0.
θ
(64)
Taking into account (63) and (64)1 , the second Piola-Kirchhoff stress may be expressed as
S = Ft
−1
−T
.
C Ee − A(θ − θ0 ) Ft
(65)
It can be shown with the aid of (62), (61) and (64)1,2 that the rate of entropy satisfies the
relation
ρ0 η̇θ = ρ0 r − Divq0 +
X
fα ξ˙α ,
(66)
α
∂ Ψ̂
is the thermodynamic driving force for martensitic variant α. By
∂ξα
assumption, phase transformation occurs when the thermodynamic driving force reaches
where fα = −
19
Version: September 6, 2011, 14:03
Multiscale model for thermomechanical response
a critical value Fc (forward transformation) or −Fc (reverse transformation). Deriving an
expression for the entropy from (64)2 and (63), and substituting it in (66), one finds that
cθ̇ = ρ0 r − Div q0 +
X
(Bθ + fα )ξ˙α − Ee ·˙CAθ .
(67)
α
With reference to (67), it is easy to see that the cumulative volumetric heat supply r̂
comprises the external heat source r, the thermoelastic rate of heating in the form Ee ·˙CAθ
P
and the heating due to phase transformation given by α (Bθ + fα )ξ̇α . In summary, the
microscopic constitutive relations for a phase-transforming solid are
Pm = Fm Ft
−1
−T
Cm Ee − Am (θ m − θ0 ) Ft
,
m
m
qm
0 = −K Grad θ
(68)
(69)
and
m
r̂ m = ρm
0 r +
X
(Bθ m + fα )ξ̇α − Ee · C˙m Am θ m ,
(70)
α
where the superscript “m” is dropped for quantities that pertain exclusively to the microscopic problem.
8.2
Numerical simulations
The two parameters of the isotropic elastic modulus used for Nitinol are as follows: The
Young’s modulus-like parameter is volume-averaged between the austenite phase (Ea =
60.0 GPa) and the martensite phase (Em = 20.0 GPa), while the Poisson ratio-like parameter is ν = 0.3 for both phases. In addition, the thermal conductivity is taken to
be isotropic for both phases and equal to 18 W/(m K) and 8.6 W/(m K) for austenite and martensite, respectively. At the microscale, the effective isotropic conductivity
Km = K m I is subsequently defined as a volume average of the preceding austenite and
martensite conductivities. Also, the thermal expansion is isotropic and equal in both phases
to αm = 11 × 10−6 /K, while the volumetric heat capacity is cm = 5.8 MJ/(m3 K). In addition, the thermodynamic constants for the martensite variants are chosen to be B = 0.75
MPa/K, θT = 271 K and Fc = 3.5 MPa, while the reference temperature is θ0 = 295 K.
8.2.1
Textured tube under tension
The simulations presented in this section are used to validate the model against experimental results on the tension of a textured polycrystalline thin-walled Nitinol tube at different
20
Version: September 6, 2011, 14:03
A. Sengupta, P. Papadopoulos and R.L. Taylor
rates of loading. Texture is incorporated in the microscale by assigning crystal orientations
to each element of the RVE using a Monte Carlo sampling method described in [9]. The
tube has an inner diameter of 3.47 mm and thickness of 0.45 mm throughout its length of
3 in. The uniaxial stress is measured as applied load divided by the undeformed area and
the strain is determined by a strain gage in the middle section of the tube. In addition, the
temperature is recorded using a sensor attached to the middle of the tube. Non-isothermal
conditions arise from tensile loading of the tube at strain rates of 10−4 /sec and 10−3 /sec.
In the simulations, the tube is taken to be entirely fixed at both ends to reproduce the
effect of the much stiffer grips. Both conduction and convection heat transfer are accounted
for in the simulations. The ends of the tube are assumed to be at the ambient temperature
θ0 due to the large grips acting as heat sinks, thus resulting in conduction of heat between
the tube and the grips. In addition, the lateral surface of the tube is exposed to ambient
conditions which results in convection heat transfer between the tube and the surrounding
air. A standard assumption is that the normal heat flux can be adequately approximated
as qn = h(θ − θ0 )n , see [21]. Here, the value of the exponent is chosen to be n = 1.25,
based on the geometry and expected temperature of the tube [21, Chapter 7], while the
convection parameter is h = 10 for θ > θ0 and h = 80 for θ ≤ θ0 .
Given the computational intensity of the proposed multiscale method, a relatively
coarse mesh with 180 eight-node brick elements is employed to model the tube in the
macroscale. This is acceptable in light of the relatively simple nature of the applied loading. Full Gaussian integration is adopted for the finite element problem at the two scales
and each Gauss point at the macroscale is comprised of an RVE containing 43 crystals for
the displacement boundary condition and 33 crystals for the periodic boundary condition.
The latter case requires a smaller RVE size owing to its faster convergence. A cluster of
24 Intel dual-core processors with 1 Gbps Ethernet connectivity is employed for the calculations. The complete calculation described below required approximately 4 hours of CPU
time on this cluster.
In Figure 4 and Figure 5, the uniaxial stress-strain responses are compared at strain
rates of 10−4 /sec and 10−3 /sec, respectively. Both the displacement-like and periodic
boundary conditions gave results that match the experiments quite well, although it can
be seen that the displacement boundary condition stresses are higher compared to those
of the periodic boundary conditions. For the case of periodic boundary conditions, Figure
6 shows a comparison of the longitudinal stress distribution for the two strain rates. It is
observed that, although the stresses are mostly uniform over the tube, there is a region
21
Version: September 6, 2011, 14:03
Multiscale model for thermomechanical response
near the two ends of the tube where the stresses are lower. Figure 7 shows a comparison
of the martensitic volume fractions at the end of loading. It is noted here that the volume
fractions for the 10−4 /sec rate are overall higher than those of the 10−3 /sec rate, since the
lower temperature attained at this lesser strain rate is more conducive to transformation.
Figure 8 and Figure 9 show the temperature history at the middle of the tube for the two
strain rates. It is concluded that both simulations yield results that are quite close to the
experimental ones and predict the maximum and minimum temperatures with reasonable
accuracy. The only deviation of the numerical results from the experimental measurements
is at a stage of the loading which includes the formation of an intermediate R-phase and
its subsequent transformation to martensite, which are not modeled here. Figure 10 shows
the distribution of temperature in the tube at the end of loading for the two strain rates.
As expected, the temperature elevation is lower for the lower strain rate, as seen from (67).
9
Conclusions
This paper introduces a finite element-based homogenization method for fully coupled
continuum thermomechanical modeling at two scales. The method employs a monolithic
treatment of the mechanical and thermal components, which is particularly attractive
for problems that exhibit strong mechanical-thermal coupling and significant energy dissipation. An efficient algorithmic formulation is proposed that relies on a single static
condensation of internal DOFs in the microscale and minimizes the data size for communication between the two scales. A simple parallel implementation of the algorithm is also
presented that transforms a serial code into a simple parallel code that distributes the microscale computations among several processors in a time- and memory-efficient manner.
Even so, it is important to emphasize that fully-coupled thermomechanical FE2 computations are quite intensive and should be used for materials that exhibit genuinely complex
thermomechanical behavior that depends crucially on the history of microstructural evolution.
Acknowledgments
The authors would like to thank Thomas W. Duerig, Alan Pelton and Aaron Kueck of
Nitinol Devices & Components (Fremont, California) for providing the experimental results
and also for helpful discussions. Also, support for AS and PP was provided by a KAUSTAEA grant, which is gratefully acknowledged.
22
Version: September 6, 2011, 14:03
A. Sengupta, P. Papadopoulos and R.L. Taylor
References
[1] J.-M. Guedes and N. Kikuchi. Preprocessing and postprocessing for materials based
on the homogenisation method with adaptive finite element methods. Comp. Meth.
Appl. Mech. Engr., 83:143–198, 1990.
[2] S. Ghosh, K. Lee, and S. Moorthy. Multiple scale analysis of heterogeneous elastic
structures using homogenisation theory and Voronoi cell finite element method. Int.
J. Sol. Struct., 32:27–62, 1995.
[3] C. Miehe, J. Schotte, and J. Schröder. Computational micro-macro transitions and
overall moduli in the analysis of polycrystals at large strains. Comp. Mater. Sci.,
16:372–382, 1999.
[4] V.G. Kouznetsova. Computational homogenization for the multi-scale analysis of
multi-phase materials. PhD thesis, Netherlands Institute of Metals Research, 2002.
[5] B. Nadler, P. Papadopoulos, and D.J. Steigmann. Multi-scale constitutive modeling
and numerical simulation of fabric material. Int. J. Sol. Struct., 43:206–221, 2006.
[6] F. Feyel and J.L. Chaboche. FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fiber SiC/Ti composite materials. Comp. Meth. Appl.
Mech. Engrg., 183:309–330, 2000.
[7] C. Miehe, J. Schotte, and M. Lambrecht. Homogenization of inelastic solid materials at
finite strains based on incremental minimization principles. application to the texture
analysis of polycrystals. J. Mech. Phys. Solids, 50:2123–2167, 2002.
[8] I. Özdemir and W.A.M. Brekelman and. M.G.D. Geers. FE2 computational homogenization for the thermo-mechanical analysis of heterogeneous solids. Comp. Meth.
Appl. Mech. Engrg., 198:602–613, 2008.
[9] A. Sengupta, P. Papadopoulos, and R.L. Taylor. Multiscale finite element modeling
of superelasticity in Nitinol polycrystals. Comp. Mech., 43:573–584, 2009.
[10] K. Terada, M. Kurumatani, T. Ushida, and N. Kikuchi. A method of two-scale thermomechanical analysis for porous solids with micro-scale heat transfer. Comp. Mech.,
46(2):269–285, 2010.
[11] H.W. Zhang, S. Zhang, J.Y. Biyi, and B.A. Schrefler. Thermo-mechanical analysis of
periodic multiphase materials by a multiscale asymptotic homogenization approach.
Int. J. Num. Meth. Engrg., 69:87–113, 2007.
[12] J. Fish and R. Fan. Mathematical homogenization of nonperiodic heterogeneous media
subjected to large deformation transient loading. Int. J. Num. Meth. Engrg., 76:1044–
1064, 2008.
23
Version: September 6, 2011, 14:03
Multiscale model for thermomechanical response
[13] R. Hill. On constitutive macro-variables for heterogeneous solids at finite strain. Proc.
R. Soc. Lond. A, 326:131–147, 1972.
[14] I. Özdemir and W.A.M. Brekelman and. M.G.D. Geers. Computational homogenization for heat conduction in heterogeneous solids. Int. J. Num. Meth. Engrg., 73:185–
204, 2008.
[15] M. Ostoja-Starzewski. Towards stochastic continuum thermodynamics. J. Non-Equil.
Therm., 27:335–348, 2002.
[16] J.L. Auriault. Effective macroscopic description for heat conduction in periodic composites. Int. J. Heat Mass Trans., 26(6):861–869, 1983.
[17] R.L. Taylor. FEAP - A Finite Element Analysis Program, User Manual. University
of California, Berkeley. http://www.ce.berkeley.edu/feap.
[18] R.L. Taylor. FEAP - A Finite Element Analysis Program, Programmer Manual. University of California, Berkeley. http://www.ce.berkeley.edu/feap.
[19] Open MPI: Open source high performance computing.
[20] A. Sengupta, P. Papadopoulos, A. Kueck, and A.R. Pelton. On phase transformation
models for thermo-mechanically coupled response of Nitinol. Comp. Mech., to appear
2011.
[21] J.P. Holman. Heat Transfer. McGraw-Hill, Inc., New York, 1990.
Appendix: Expressions for the macroscopic referential thermomechanical tangent moduli
Expressions for the individual terms of the macroscopic referential thermomechanical tangent moduli derived earlier in Section 5.3 are elaborated here in context of the unified
computational representation of Section 7.1.
The mechanical tangent modulus can be shown to be the [{1:9},{1:9}] submatrix of
[Ă] computed using (55), by rewriting (23) as
[
1 X X
∂P M
EF
[X E ][Ĉuu
][X F ] =
]
=
∂F M
V
E∈ext F ∈ext
"
#
X X X X
1
b
E
EI
JI −1
JF
F
[D] −
[X ] [C ][C ] [C ] uu [X ] . (A.1)
V
I∈int J∈int E∈ext F ∈ext
24
Version: September 6, 2011, 14:03
A. Sengupta, P. Papadopoulos and R.L. Taylor
Likewise, the tangent moduli due to thermal expansion can also be shown to be the
[{1:9},10] submatrix of [Ă], by rewriting (33) as
[
1 X X
∂P M
EF
][1F ] =
[X E ][Ĉuθ
] =
M
∂θ
V
E∈ext F ∈ext
"
#
X X X X
1
[DA]b −
[X E ] [C EI ][C JI ]−1 [C JF ] uθ [1F ] . (A.2)
V
I∈int J∈int E∈ext F ∈ext
Next, the tangent for the cumulative heat supply r̂ M with respect to deformation given by
(41) can also be rewritten as below to show that it is the [10,{1:9}] submatrix of [Ă]:
"
#
X X X
1
∂r̂ M
I
[rF ]b −
[rF
] [C JI ]−1 [C JE ] uu [X E ] .
(A.3)
[ M] =
∂F
V
I∈int J∈int E∈ext
Similarly, the tangent for the cumulative heat supply r̂ M with respect to temperature given
by (42) can be rewritten to show that it is the [10,10] submatrix of [Ă] as follows:
[
∂r̂ M
1 h b
[rθ ] −
]
=
∂θ M
V
X X X
I∈int J∈int E∈ext
I
[rθI ] [C JI ]−1 [C JE ] θθ [1E ] − [rF
] [C JI ]−1 [C JE ] uθ [1E ]
#
. (A.4)
Finally, the tangent thermal conductivity can be shown to be the [{11:13},{11:13}] submatrix of [Ă], by rewriting (36) as
"
#
X X X X
1
b
E
EI
JI
−1
JF
F
c
[K] −
[X ] [C ][C ] [C ] θθ [X − X ] . (A.5)
[K M ] =
V
I∈int J∈int E∈ext F ∈ext
25
Version: September 6, 2011, 14:03
Multiscale model for thermomechanical response
+
+
_
_
Figure 1: Deformation of an RVE under periodic boundary conditions
26
Version: September 6, 2011, 14:03
A. Sengupta, P. Papadopoulos and R.L. Taylor
EI ]
[X E ][Cuθ
EI ]
[X E ][Cuu
[D]b
[G̃J ]
JF ][X F − X c ]
[Cuθ
JF ][X F − X c ]
[Cθθ
JI ]
[Cθθ
θ M Grad θ M
(1)
(3)
JF ][1F ]
[Cuθ
JF ][X F ]
[Cuu
JI ]
[Cθu
JI ]
[Cuθ
JF ][X F ]
[Cθu
uJ
(3*Nint)
θJ
(Nint)
FM
(9)
JI ]
[Cuu
FM
(9)
JF ][1F ]
[Cθθ
θI
(Nint)
uI
(3*Nint)
M
∂P
]
[DA][b∂(Grad
θ)M
Grad θ M θ M
(3)
(1)
[H]
[rθI ]
I]
[rF
[rF
]b
∂qM
EI ]
[X E ][Cθu
EI ]
[X E ][Cθθ
[ ∂F0M ]
[rθ
]b[
∂qM
0
[ ∂θM
]
∂ r̂
]
∂(Grad θ)M
[K]b
[GI ]
Figure 2: Matrices and their entries required for computing the homogenized coupled tangents
27
Version: September 6, 2011, 14:03
Multiscale model for thermomechanical response
MPI Start
2
1
0
p
i
MPI Send
input
FM , θ M ,
Grad θ M
retrieve/
solve
store
RVE
model
state
compute
MPI Recv
PM , qM
0 , [Ă]
Figure 3: Schematic of the MPI implementation for p + 1 processors
28
Version: September 6, 2011, 14:03
A. Sengupta, P. Papadopoulos and R.L. Taylor
Uniaxial stress (MPa)
600
500
400
300
200
Experiment
Periodic b.c.
Displacement b.c.
100
0
0
1
2
3
4
5
Uniaxial Strain
6
Figure 4: Comparison of stress response under longitudinal tension at 10−4 /sec strain rate
Uniaxial Stress (MPa)
700
600
500
400
300
200
Experiment
Periodic b.c.
Displacement b.c.
100
0
0
1
2
3
Uniaxial Strain
4
5
6
Figure 5: Comparison of stress response under longitudinal tension at 10−3 /sec strain rate
29
Version: September 6, 2011, 14:03
Multiscale model for thermomechanical response
_________________
STRESS 3
_________________
STRESS 3
4.65E+08
5.29E+08
4.97E+08
5.29E+08
5.70E+08
6.11E+08
5.61E+08
6.52E+08
5.93E+08
6.24E+08
6.93E+08
7.33E+08
6.56E+08
7.74E+08
6.88E+08
7.20E+08
8.15E+08
8.56E+08
7.52E+08
8.97E+08
7.83E+08
8.15E+08
9.38E+08
9.79E+08
8.47E+08
1.02E+09
Time = 1.10E+03
Time = 1.10E+02
Figure 6: Longitudinal stress distributions for 10−4 /sec and 10−3 /sec strain rates
____________________
MARTENSITE FRCN.
____________________
MARTENSITE FRCN.
1.60E-01
2.43E-01
2.09E-01
2.57E-01
2.84E-01
3.25E-01
3.05E-01
3.66E-01
3.53E-01
4.02E-01
4.06E-01
4.47E-01
4.50E-01
4.88E-01
4.98E-01
5.47E-01
5.29E-01
5.69E-01
5.95E-01
6.10E-01
6.43E-01
6.91E-01
6.51E-01
6.92E-01
7.40E-01
7.32E-01
Time = 1.10E+03
Time = 1.10E+02
Figure 7: Martensite volume fractions for 10−4 /sec and 10−3 /sec strain rates
30
Version: September 6, 2011, 14:03
A. Sengupta, P. Papadopoulos and R.L. Taylor
350
340
330
Temperature (K)
320
310
300
290
280
Experiment
Periodic b.c.
Displacement b.c.
270
260
250
0
500
1000
Time
1500
2000
2500
Figure 8: Comparison of temperature history at middle of tube at 10−4 /sec strain rate
350
340
330
Temperature (K)
320
310
300
290
280
Experiment
Periodic b.c.
Displacement b.c.
270
260
250
0
50
100
Time
150
200
250
Figure 9: Comparison of temperature history at middle of tube at 10−3 /sec strain rate
31
Version: September 6, 2011, 14:03
Multiscale model for thermomechanical response
_________________
TEMPERATURE
_________________
TEMPERATURE
0.00E+00
0.00E+00
2.90E+02
2.92E+02
2.90E+02
2.92E+02
2.94E+02
2.94E+02
2.96E+02
2.98E+02
2.96E+02
2.98E+02
3.00E+02
3.00E+02
3.02E+02
3.04E+02
3.02E+02
3.04E+02
3.06E+02
3.06E+02
3.08E+02
3.10E+02
3.08E+02
3.10E+02
3.01E+02
3.10E+02
Time = 1.10E+03
Time = 1.10E+02
Figure 10: Temperature distributions for 10−4 /sec and 10−3 /sec strain rates
32
Version: September 6, 2011, 14:03