mathematical homogenization of nonperiodic heterogeneous media subjected to large deformation...

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2008; 76:1044–1064Published online 19 June 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2355

Mathematical homogenization of nonperiodic heterogeneousmedia subjected to large deformation transient loading

Jacob Fish∗,† and Rong Fan

Multiscale Science and Engineering Center, Rensselaer Polytechnic Institute, Troy, NY 12180, U.S.A.

SUMMARY

We present a generalization of the classical mathematical homogenization theory aimed at accountingfor finite unit cell distortions, which gives rise to a nonperiodic asymptotic expansion. We introducean auxiliary macro-deformed configuration, where the overall Cauchy stress is defined, and nonperiodicboundary conditions. Verification studies against a direct numerical simulation demonstrate the versatilityof the proposed method. Copyright q 2008 John Wiley & Sons, Ltd.

Received 2 February 2008; Revised 20 February 2008; Accepted 25 February 2008

KEY WORDS: multiscale; mathematical homogenization; nonperiodic; unit cell; macro; micro; largedeformation

1. INTRODUCTION

This paper presents a mathematical homogenization framework for large deformation of a nonpe-riodic heterogeneous medium subjected to transient loading. The subject of mathematical homoge-nization dates back to the pioneering works of Babuska [1], Benssousan et al. [2], Sanchez-Palencia[3] and Bakhvalov and Panassenko [4]. Engineering homogenization approaches date back to Hill[5]. After more than 30 years and over 734 000 Google hits on the word ‘homogenization’ thisarticle contribution is on the following two aspects:

(i) Theory: There is a common belief [6] that methods based on the multiple scale asymptoticexpansion [1–4] are limited to small deformation problems and this leads to so-called computationalhomogenization approaches, which are based on the Hill–Mandel relation [7], in combination with

∗Correspondence to: Jacob Fish, Multiscale Science and Engineering Center, Rensselaer Polytechnic Institute, Troy,NY 12180, U.S.A.

†E-mail: [email protected]

Contract/grant sponsor: Office of Naval Research; contract/grant number: N000140310396Contract/grant sponsor: Air Force Research Laboratory; contract/grant number: 478-05-506E

Copyright q 2008 John Wiley & Sons, Ltd.

NONPERIODIC HETEROGENEOUS MEDIA 1045

numerical methods (see [8–20]) and for closely related homogenization-like methods [21–26].Although mathematical homogenization in combination with finite elements has been successfullyemployed to capture material non-linearities ([27–30] including large unit cell rotations [31, 32]),the inherent assumption of asymptotic methods that macroscopic fields remain constant over aunit cell domain is questionable at best for large cells undergoing significant distortion. Thepresent article presents a generalization of the mathematical homogenization theory that permitsconsideration of large unit cells undergoing significant nonperiodic distortion.

(ii) Practice: Wave propagation in a medium with embedded local weak and/or strong disconti-nuities is one of the most attractive applications of the method. Provided that the usual assumptionabout the smallness of the local problem holds, the original initial boundary value problem canbe decomposed into macro and micro problems with the micro problem being a steady-state (orstatic) problem. This avoids the use of small finite elements in the macro problem, which governthe time step of explicit calculations.

This manuscript is organized as follows. Generalization of the mathematical homogenizationtheory to account for large nonperiodic unit cell distortions and its computational implementationare described in Section 2. Verification studies against a direct numerical simulation where localdetails are computationally resolved are conducted in Section 3.

2. MATHEMATICAL HOMOGENIZATION FOR GEOMETRICALLYNON-LINEAR PROBLEMS

In deriving the coupled two-scale problem, the governing equations at the smallest scale of interest(microscale) will be stated in terms of the deformation gradient and the first Piola–Kirchhoff stress.This pair of work conjugate variables has been originally advocated by Hill [7] and later adopted in[10, 18, 20]. The choice of the deformation gradient (and its conjugate stress measure) is convenientin defining the essential boundary conditions (BCs) on the unit cell. In the discretization phase,the finite element equations will be reformulated in terms of Cauchy stress for problems whereconstitutive equations at the microscale are defined in terms of Cauchy stress.

The strong form of the initial boundary value problem on domain �X with boundary �X isgiven as

�P�i j (F

�)

�X j+b�

Xi−��X

�2u�i

�t2= 0 on �X (1)

F�ik = �ik+ �u�

i

�Xk(2)

with BCs

P�i j NX j = t Xi on �t

X (3)

u�i = ui on �u

X (4)

�tX ∪�u

X =�X and �tX ∩�u

X =0 (5)

Copyright q 2008 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2008; 76:1044–1064DOI: 10.1002/nme

1046 J. FISH AND R. FAN

and initial conditions

u�i = u0i at t=0 (6)

�u�i

�t= v0i at t=0 (7)

where P� denotes the first Piola–Kirchhoff stress tensor, F� the deformation gradient, bX the bodyforce vector, NX the unit normal to the boundary �X and �X the initial mass density. Lowercase subscripts i and j denote spatial dimensions. Subscripts X and x refer to the initial anddeformed configurations, respectively. The superscript � denotes dependence of a variable on thefine (micro)-scale features. X is a material coordinate in the initial macroscopic domain �X and x isthe corresponding deformed (spatial) coordinate. Summation convention over repeated subscriptsis employed except for subscripts X and x .

2.1. Asymptotic expansion for geometrically non-linear problems

In the classical mathematical homogenization theory, displacements are expanded as

u�i (X, t)=u0i (X, t)+�u1i (X,Y, t)+�2u2i (X,Y, t)+O(�3) (8)

where Y represents material coordinates in the initial microscopic (unit cell) domain �Y and y isthe deformed (spatial) coordinate in the unit cell domain �y . The two scale coordinates are relatedby Y≡X/� with 0<��1.

In the classical theory, it is assumed that the size of the unit cell is of the order �, i.e. infinitesimallysmall and therefore the macroscopic displacement u0i (X, t) is considered to be constant over theunit cell domain. For large unit cell distortions, the macro displacement u0i (X, t) is no longerconstant over the unit cell domain; therefore, Equation (8) has to be modified as described below.

We start by expanding u0i (X, t) in Taylor’s series around an arbitrary reference point X= X inthe unit cell (see Figure 1)

u0i (X, t)=u0i (X, t)+ �u0i�X j

∣∣∣∣∣X

(X j− X j )+ 1

2

�2u0i�X j�Xk

∣∣∣∣∣X

(X j− X j )(Xk− Xk)+·· · (9)

As X j− X j=�Y j , we can write

u0i (X, t)=u0i (X, t)+ ��u0i�X j

∣∣∣∣∣X

Y j+�21

2

�2u0i�X j�Xk

∣∣∣∣∣X

Y jYk+O(�3) (10)

Similarly, the higher-order terms uni (X,Y, t) for n�1 in (8) are expanded around X= X as

uni (X,Y, t)= uni (X,Y, t)+ �uni�X j

∣∣∣∣X

(X j− X j )+ 1

2

�2uni�X j�Xk

∣∣∣∣∣X

(X j− X j )(Xk− Xk)+·· ·

= uni (X,Y, t)+ ��uni�X j

∣∣∣∣XY j+�2

1

2

�2uni�X j�Xk

∣∣∣∣∣X

Y jYk+O(�3) (11)



y1

Initial

Unit Cell

Y1

Y2

y2

1 1X , x

2 2X , x

O

O

O

Deformed

Unit Cell

ˆ=X X ˆ=X x

Figure 1. Macro and micro coordinate systems.

Substituting (10) and (11) into (8) gives

u�i (X, t)=ui (X,Y, t)= u0i (X, t)+�u1i (X,Y, t)+�2u2i (X,Y, t)+O(�3) (12)

where

u0i (X, t)= u0i (X, t) (13)

u1i (X,Y, t)= u1i (X,Y, t)+ �u0i�X j

∣∣∣∣∣X

Y j (14)

u2i (X,Y, t)= u2i (X,Y, t)+ �u1i�X j

∣∣∣∣∣X

Y j+ 1

2

�2u0i�X j�Xk

∣∣∣∣∣X

Y jYk (15)

In classical homogenization, the spatial derivative is defined as

� f �

�Xi= � f (X,Y, t)

�Xi+ 1

�

� f (X,Y, t)

�Yi(16)

In the case f (X,Y, t) rather than f (X,Y, t), the spatial derivative takes the form

� f �(X, t)

�Xi= 1

�

� f (X,Y, t)

�Yi(17)

The displacement gradient components may be expressed as

�u�i

�Xk= 1

�

�ui (X,Y, t)

�Yk= �u1i (X,Y, t)

�Yk+�

�u2i (X,Y, t)

�Yk+O(�2) (18)



where

�u1i (X,Y, t)

�Yk= �u1i (X,Y, t)

�Yk+ �u0i (X, t)

�Xk

∣∣∣∣∣X

(19)

�u2i (X,Y, t)

�Yk= �u2i (X,Y, t)

�Yk+ �u1i (X,Y, t)

�Xk

∣∣∣∣∣X

+ �2u1i (X,Y, t)

�X j�Yk

∣∣∣∣∣X

Y j+ �2u0i�X j�Xk

∣∣∣∣∣X

Y j (20)

The deformation gradient can be expressed as

F�ik=�ik+ �u�

i

�Xk=F0

ik(X,Y, t)+�F1ik(X,Y, t)+O(�2) (21)

where �i j is the Kronecker delta and

F0ik(X,Y, t)= �ik+ �u1i (X,Y, t)

�Yk≡Fik(X,Y, t) (22)

F1ik(X,Y, t)= �u2i (X,Y, t)

�Yk(23)

The overall (macro) deformation gradient is obtained by integrating the leading-order termFik(X,Y, t) in (22) over the unit cell domain, which yields

Fik(X, t)= 1

|�Y |∫

�Y

Fik(X,Y, t)d�Y =�ik+ �u0i (X, t)

�Xk

∣∣∣∣∣X

(24)

where for the time being (see Section 2.2 for the nonperiodic case) we assume Y-periodicity ofu1i (X,Y, t). A function f is considered to be periodic if f (X,Y+kL)= f (X,Y), where L is acharacteristic size of the unit cell, i.e. f (X,Y) is equal on the opposite of a unit cell.

The first Piola–Kirchhoff stress Pi j (F(X,Y, t)) formulation is adopted because of its conjugacyto the deformation gradient. If we assume a two-term asymptotic expansion in (12) (for additionaldiscussion, see Appendix) and expanding Pi j (F(X,Y, t)) in Taylor’s series around X= X yields

Pi j (F(X,Y, t))= Pi j (F(X,Y, t))+ �Pi j�Xk

∣∣∣∣X

(Xk− Xk)+·· ·

= Pi j (F(X,Y, t))+��Pi j�Xk

∣∣∣∣XYk+O(�2) (25)

Inserting the asymptotic expansions (25) into the equation of motion (1) and exploiting the definitionof the spatial derivative (17) give

�−1�Pi j�Y j+ �Pi j

�X j+bXi (X,Y)−�X (X,Y)

�2u0i (X, t)

�t2+O(�)=0 (26)



Identifying terms with equal powers of � yields the following two leading-order momentumequations:

O(�−1) : �Pi j�Y j=0 (27)

O(�0) : �Pi j�X j+bXi (X,Y)−�X (X,Y)

�2u0i (X, t)

�t2=0 (28)

Integrating (28) over the unit cell domain gives the macroscopic equation of motion

�Pi j (X, t)

�X j+bXi (X)−�X (X)

�2u0i (X, t)

�t2=0 (29)

where

Pi j (X, t)= 1

|�Y |∫

�Y

Pi j (F(X,Y, t))d�Y (30)

bXi (X)= 1

|�Y |∫

�Y

bXi (X,Y)d�Y , �X (X)= 1

|�Y |∫

�Y

�X (X,Y)d�Y (31)

Remark 1Note that even if the original asymptotic expansion (8) were periodic in Y, then its modifiedversion (12) would not be periodic due to its dependence on Y j . Equation (24) was the onlyinstance where the periodicity assumption was made. In the following section, we will explorevarious possibilities on how to satisfy Equation (24) without the periodicity assumption.

2.2. BCs for the unit cell problem

In this section, we will consider various BCs for the unit cell problem starting with the classicalperiodic case.

Consider an asymptotic expansion of the displacement field (12)

u�i (X, t)=ui (X,Y, t)=u0i (X, t)+� [(Fi j (X, t)−�i j )Y j+u1i (X,Y, t)]︸︷︷︸

u1i (X,Y,t)

+O(�2) (32)

where we have accounted for Equation (24).The leading-order term in (32) represents the rigid body translation of the unit cell u0i (X, t),

which is independent of the unit cell coordinates Y. The unit cell distortion is captured by twoO(�) terms. The first term, (Fi j (X, t)−�i j )Y j , represents a uniform macroscopic deformation,whereas the second term, u1i (X,Y, t), captures the deviation from the uniform field introduced bymaterial heterogeneity.

Figures 2(a) and (b) show the initial and deformed shape of the unit cell, respectively. Thedotted line in Figure 2(b) depicts the deformed shape of the unit cell due to (Fi j (X, t)−�i j )Y j ,whereas the solid line shows the contribution of the two O(�) terms.

At the unit cell vertices, �vertY , the deviation from the uniform field, u1i (X,Y, t), is assumed to

vanish. For the remaining points on the boundary of the unit cell, the deviation from the average



(b)(a) Y1

Y2

M S

1C

3C 4C

2C

YΘ

1y

M ′ M ′′ S ′S′′

2C ′

1C ′

3C ′ 4C ′2y

yΘ

Figure 2. Definition of periodic boundary conditions: (a) initial unit cell and (b) deformed unit cell.

u1i (X,Y, t) is prescribed to be a periodic function. Figure 2 depicts two points M and S on theopposite faces of the unit cell with M and S being master and slave points, respectively. Thedisplacement of the two points is given as

u1i (X,YM, t)= (Fi j (X, t)−�i j )YMj +u1i (X,YM, t) (33)

u1i (X,YS, t)= (Fi j (X, t)−�i j )YSj +u1i (X,YS, t) (34)

Subtracting Equation (34) from (33) and accounting for periodicity, u1i (X,YM, t)=u1i (X,YS, t),we get the periodic BC

u1i (X,YM, t)− u1i (X,YS, t)=(Fi j (X, t)−�i j )(YMj −Y S

j ) (35)

where YM and YS represent the coordinates of the master and slave nodes on the unit cell boundary,respectively.

As noted in Remark 1, the only instance where we assumed periodicity was in deriving Equation(24). For a nonperiodic medium, we will now re-examine the derivation of (24). For (24) to hold,the following condition must be satisfied:

∫�Y

�u1i (X,Y, t)

�Y jd�Y =0

Applying Green’s theorem and exploiting relations (14) and (24) the above reduces to

∫�Y

(u1i (X,Y, t)−(Fik(X, t)−�ik)Yk)NY j d�Y =0 (36)

where �Y is the boundary of�Y and NY j are the components of the unit normal to the boundary �Y .Alternatively to Equations (35) and (36), an essential BC

u1i (X,Y, t)−(Fik(X, t)−�ik)Yk=0 (37)

is often exercised in practice.



(a) 1C′ 1y

2C′

3C′4C′

2y

yΘ

1C′ 1y

2C′

3C′4C′

2y

yΘ

2C′

1C′

3C′ 4C′

1y

yΘ

2y

(b) (c)

Figure 3. Definition of (a) essential, (b) natural and (c) mixed boundary conditions.

The essential BC (37) can be enforced in the weak form as∫�Y

(u1i (X,Y, t)−(Fik(X, t)−�ik)Yk)�i d�Y =0 (38)

where �i is a Lagrange multiplier representing unknown tractions on �Y . If we choose �i = Pi j NY j

with Pi j being constant over �Y and require (38) to be satisfied for arbitrary Pi j , we then obtainEquation (36). We will refer to Equation (36) as a natural BC. Equation (36) is in the spirit of theBC proposed by Mesarovic and Padbidri [33] who required the unit cell to satisfy average smallstrains.

The essential (37) and natural (36) BCs can be combined in the so-called mixed BC as follows:

∫�Y

(u1i (X,Y, t)−(Fik(X, t)−�ik)Yk)NY j d�Y =0

|(u1i (X,Y, t)−(Fik(X, t)−�ik)Yk)NY j |��

(39)

Note that the mixed BC (39) is more restrictive than the natural BC (36) but is more compliantthan the essential BC (37), which is enforced up to a tolerance �. The three types of unit cellBCs are schematically illustrated in Figure 3. The mixed BC can be implemented in several ways:perhaps the simplest is by defining double nodes on the boundary and placing a linear or non-linearspring between them as illustrated in Figure 3(c). We will discuss the definition of spring stiffnessin the numerical examples section. Note that in the case of a very stiff spring the mixed BC (39)coincides with the essential BC (37); for an infinitely compliant spring it reduces to the naturalBC (36).

We will examine the four types of BCs for their suitability for large deformation problems. Wewill denote the four BCs as

g(u1)=0 on �Y (40)

2.3. The two-scale problem

The two-scale problem derived in the previous sections is two-way-coupled. The link between thetwo scales is schematically shown in Figure 4. The fine-scale problem is driven by the overall



Fine scale problem Coarse scale problem

0 ˆ( , )ˆ( , )ˆ

iik ik

k

u tF t

Xδ ∂= +

∂X

X

( )( )1ˆ ˆ( , ) , ,Y

YjijiY

P t P t dΘ

Θ=Θ ∫X F X Y

Figure 4. Information transfer between the macro and micro problems.

(coarse scale) deformation gradient Fik(X, t). It informs the coarse-scale problem about the overallstress Pi j (X, t). The coupled two-scale problem is summarized below:

(a) Fine-scale problem:

Given Fik(X, t), find u1i (X,Y, t) on �Y such that

�Pi j (F(X,Y, t))

�Y j=0 on �Y

g(u1)=0 on �Y

(41)

(b) Coarse-scale problem:

Given Pi j , find u0i (X, t) on �X such that

�Pi j

�X j+bXi =�X

�2u0i�t2

on �X

Pi j NX j = t Xi on �tX ; u0i =ui on �u

X

u0i =u0i at t=0; �u0i�t=v0i at t=0

(42)

2.4. Finite element discretization

Both the coarse- and fine-scale problems are solved using finite elements.The displacement of the fine-scale problem u1i (X,Y, t) is approximated as

u1i (X,Y, t)=N 1B(Y)d1i B(X, t) (43)

where subscript B denotes the node number, N 1B(Y) are the unit cell shape functions and d1i B are

the nodal displacements.Let dMjC (X, t) be the master (independent) degrees and express d1i B by a linear combination of

dMjC (X, t) denoted as d1i B(X, t)=Ti BkC (X, t)dMkC (X, t) so that the constraint equation (40) in the



discrete form is satisfied

g(N 1B(Y)Ti B jCd

MjC (X, t))=0 on �Y (44)

Then writing the Galerkin weak form of (41) and discretizing it using (43) yield the discreteresidual equation

r1kC (m+1n+1 �d1)≡∫

�Y

Ti BkC�N 1

B

�Y j

m+1n+1 Pji d�Y =0 (45)

where the left subscript and superscript denote the load increment and the iteration count (forthe implicit method) at the coarse scale, respectively; m+1

n+1 r1i B and m+1n+1 �d1i B are the residual and

displacement increments in the (m+1)th iteration of the (n+1)th load increment, respectively. Ifthe constitutive equations are defined in terms of the Cauchy stress, it is convenient to restate theunit cell problem as follows:

Given m+1n+1 Fi j and n�i j , find m+1

n+1 �d1i B such that

r1kC (m+1n+1 �d1)≡∫

�y

Ti BkC�N 1

B

�y jm+1n+1 � j i d�y=0

d1i B(X, t)=Ti BkC (X, t)dMkC (X, t)

(46)

where we have exploited the relationship between the first Piola–Kirchhoff stress Pi j and theCauchy stress � j i

J� j i=Fjk Pik (47)

and J in (47) is the determinant of Fjk .Similarly, the coarse-scale displacements u0i (X, t) are discretized as

u0i (X, t)=N 0A(X)d0i A(t) (48)

where N 0A(X) and d0i A are the coarse-scale shape functions and nodal displacements, respectively.

Writing the weak form of (42) and using discretization (48) the discrete coarse-scale equationscan be expressed as

Given n+1bi and n+1t i , find n+1�d0i A such that

n+1r0i A(n+1�d0)≡Mi j ABn+1d0j B+n+1 f inti A −n+1 f exti A =0n+1d0i A=n+1ui A on �u

X

n←n+1, Go to the next load increment

(49)



where n+1r0i A and n+1�d0i A are the coarse-scale residual and displacement increments in the(n+1)th load increment, respectively, and

Mi j AB = �i j

∫�X

�X N0AN

0B d�X (50)

f inti A =∫

�X

�N 0A

�X jP ji d�X (51)

f exti A =∫

�X

N 0AbXi d�X+

∫�tX

N 0At Xi d�X (52)

where Mi j AB is the mass, f inti A and f exti A are the internal and external forces, respectively. It is againconvenient to express the coarse-scale internal and external forces in the deformed configuration as

f inti A =∫

�x

�N 0A

�x j� j i d�x (53)

f exti A =∫

�x

N 0Abxi d�x+

∫�tx

N 0At xi d�x (54)

where

�x = �X/J , bxi=bXi/J , t xi d�x= t Xi d�X (55)

� j i = F jk Pik/J (56)

and J is the determinant of Fi j .We now focus on deriving a closed-form expression for the overall Cauchy stress � j i . Substituting

(47) into (30) and recalling d�y= J d�Y , we have

Pik(X, t)= 1

|�Y |∫

�y

F−1km �mi d�y (57)

Inserting (57) into (56) and denoting the volume of the macro-deformed configuration as |�∗y |=J |�Y |, the overall Cauchy stress can be expressed as

� j i= 1

|�∗y |∫

�y

�F−1jm �mi d�y (58)

where �F−1jm =F jk F−1km maps the micro-deformed configuration �y into the macro-deformed

configuration �∗y as illustrated in Figure 5.The macro problem may be solved using either explicit or implicit time integration [34]. For the

implicit method, consistent tangent stiffness at every material point can be computed numerically(see [31, 34, 35] for details).



Y1

Y2

(a)

1C

3C 4C

2C

1C′1y

2C′

1C′

3C′ 4C′

*1y

*2y

(b)

2C′

3C′ 4C′2y

(c)

F

∆F

F

YΘ

*yΘyΘ

Figure 5. Unit cell configurations: (a) initial; (b) macro-deformed(intermediate); and (c) micro-deformed (final).

3. NUMERICAL EXAMPLES

To verify the proposed formulation we consider three test problems: a perforated plate, a platewith a centered hole and a delamination of a sandwich shell subjected to impact loading.

3.1. Perforated plate

The geometric configuration of the perforated plate is shown in Figure 6(a). The width W andthe length L of the plate are 60.0mm. Circular holes of radius 1mm are uniformly distributedin a rectangular arrangement. The right side of the plate is fixed; the top and bottom sides arefree, and a constant velocity V =6m/s is applied to the left side for the duration of 3ms. Planestrain condition is assumed. The material is assumed to be hyperelastic. The strain energy isexpressed in neo-Hookean form with initial shear modulus of 160MPa and bulk modulus equalto 4000MPa. The mass density of the material is 1.14×103 kg/m3. For multiscale analysis, themacro-scale mesh contains 25 four-node reduced integration quadrilateral elements; all elementsare 12mm × 12mm. The size of the unit cell is 4×4mm as shown in Figure 6(b). The referencesolution is obtained by direct finite element discretization of the plate with a sufficient number ofelements.

The reaction force histories obtained with the multiscale method and the reference solutionin X and Y directions are shown in Figure 7. It can be seen that the multiscale method is ingood agreement with the reference solution. Figure 8(a) depicts the von Mises stress in the regionaround the center hole at t=3ms as obtained with the multiscale solution. Figure 8(b) comparesthe multiscale method with a reference solution in the same subdomain and time instance. Inthis example, the time step for the multiscale analysis is 30 times larger than for the referencesolution.



V

X

Y

(b)

(a)

Figure 6. Geometry, boundary conditions and unit cell discretization: (a) macro-geometry and boundaryconditions and (b) mesh of the unit cell.

Figure 7. Reaction force in x (top) and y (bottom) directions.



Figure 8. Unit cell deformation and von Mises stress obtained fromthe (a) multiscale and (b) reference solutions.

V

Figure 9. Plate with a hole in the center.

3.2. Plate with a center hole

Geometry and BCs of the plate with a center hole are schematically shown in Figure 9. The widthW and the length L of the plate are 100.0mm. A circular hole of radius r=1.0mm is placed atthe center. Plane strain condition is assumed and the material properties are the same as in thefirst example. A constant velocity V =50m/s is applied to the left side for a period of 0.3ms.The right side of the plate is constrained in the horizontal direction. For the multiscale analysis,the size of the unit cell is 4×4mm and the corresponding coarse mesh element is located at thecenter of the plate as shown by the dashed lines in Figure 9. The unit cell mesh is shown inFigure 6(b). Figure 10 depicts the deformed configuration and von Mises stress in the unit cell at0.3ms as obtained by different methods: (a) reference solution by direct finite element method,(b) multiscale solution with essential BCs (37), (c) multiscale solution with natural BCs (36) and(d) multiscale solution with mixed BCs (39).

For the mixed BC, the linear spring stiffness (see Figure 3(c)) has been a priori determinedas follows. At every node (one at a time) in the neighboring subdomain domain to the unit cell(without a hole) we prescribe a small displacement in the direction normal to the boundary andcalculate the reaction force at that node. Assuming linear spring stiffness the reaction divided bythe prescribed displacement is equal to the spring stiffness at that node. Similarly, a non-linearspring stiffness (not considered in this article) can be a priori defined.



It can be seen that the essential BC is not equipped with the necessary kinematics to providethe correct deformation on the unit cell boundary. The natural BC has the right kinematics but istoo flexible. Finally, the mixed BC seems to be able to capture the overall deformation correctly.The periodic BC was not considered in this case as the deformation (see Figure 10(a)) is clearlynot periodic.

3.3. Sandwich shell

The geometry, layup and impact loading conditions are shown in Figure 11. The impactor ismodeled as a rigid body with the initial velocity of 275m/s and mass equal to 145 g. The circularplate has a diameter of 154mm. It consists of two layers, DH36 steel and polyurea, glued byepoxy. The plate is discretized with 8-node reduced integration and hourglass control [36, 37] brickelements. The DH-36 structural steel is modeled as a thermo-plastic material with von Mises yieldsurface proposed by Nemat-Nasser and Guo [38]. As the impact process involves large plasticdeformation in less than 1ms, temperature increases due to plastic deformation. Adiabatic andisothermal conditions are assumed for the DH36 steel and polyurea, respectively. The polyureais modeled using the pressure- and temperature-dependent hyperelastic–viscoelastic constitutivelaw [39] where hyperelastic effects are modeled using the Mooney–Rivlin model, whereas viscous

Figure 10. Comparison of the unit cell deformation and von Mises stress at 0.3ms as obtainedwith (a) reference solution by direct finite element method; (b) multiscale solution with essentialboundary conditions; (c) multiscale solution with natural boundary conditions; and (d) multiscale

solution with mixed boundary conditions.

V=275m/s

DH36 steel plate

Thickness=5.0mm

Epoxy interface

Polyurea plate

Thickness=5.8mm

Figure 11. Geometry, layup and impact loading of the sandwich shell.



(a)

(b)

PolyureaDH36 RVE

DH36 thickness=0.2mm

Polyurea, thickness=0.18mm

Epoxy,

,

thickness=0.02mm

B

A

C

Figure 12. Initial configuration of the (a) undeformed coarse-scale model (macro)and (b) undeformed unit cell models.

Figure 13. Residual velocity of the impactor.

effects are represented by Prony series. The epoxy is modeled using cohesive elements proposedby Liechti and Wu [40].

Figure 12 displays the geometrical configuration for the multiscale analysis. The interface anda thin layer of steel and polyurea above and below it define a unit cell (or representative volumeelement) as shown in Figure 12. The residual velocity of the impactor obtained with the multiscalemethod and direct finite element simulation (reference solution) is compared in Figure 13. Thevon Mises stress history at point C (in Figure 12(a)) is shown in Figure 14 for both the referenceand multiscale solutions. Figure 15 compares the overall deformation of the sandwich plate att=15ms as obtained with the reference and multiscale solutions. Figure 16 compares the unit celldeformation in the two locations around the interface marked in Figure 15(b) as obtained with



Figure 14. von Mises stress at point C (in Figure 12(a)).

(a)Debonding

(b)Region a Region b

Figure 15. Deformation of the sandwich plate obtained with the (a) reference and(b) multiscale solutions at t=15ms.

the reference and multiscale solutions. The location of points A and B prior to the deformation isshown in Figure 12(b); the position of the two points at t=15ms is depicted in Figure 16 at thetwo macroscopic positions denoted as regions a and b in Figure 15(b).

4. CONCLUSIONS AND FUTURE RESEARCH DIRECTIONS

The mathematical homogenization theory has been generalized to account for finite nonperiodicdeformation of heterogeneous medium subjected to transient loading. An auxiliary macro-deformedconfiguration, where the overall Cauchy stress is updated, and mixed unit cell BCs for modelingnonperiodic deformation were introduced and verified.

In all verification problems considered, the overall deformation as obtained with the multiscalemethod has been found to be in good agreement with the reference solution; the error in the



(a) Reference solution

1.11mm

1.92mm

A

B

Multiscale solution

1.76mm

0.98mm

A

B

(b) Reference solution

2.65mm

2.81mm

A

B

Multiscale solution

2.59mm

2.48mm

B

A

Figure 16. Local deformation in (a) region a and (b) region b (see Figure 15(b)) at t=15msas obtained with reference (left) and multiscale (right) solutions.

localized fields was often as high as 10% (sandwich plate problem) but the time integration stepwas reduced by a factor of 30 compared with the reference solution.

Future work will focus on the following two issues:

(a) model reduction to reduce the cost of unit cell calculations and(b) alternative formulation of mixed boundary condition that would not require a priori cali-

bration.

APPENDIX A

In the appendix we consider an implication of considering a complete asymptotic expansion (12)as opposed to just the first two terms considered in the text. Expanding Pi j (F�(X, t)) in Taylor’sseries around X= X yields

Pi j (F�(X, t))= Pi j (F�(X, t))+ �Pi j�Xk

∣∣∣∣X

(Xk− Xk)+·· ·

= Pi j (F�(X, t))+��Pi j�Xk

∣∣∣∣XYk+O(�2) (A1)

The leading-order stress Pi j (F�(X, t)) can be expanded around the leading-order deformationgradient F(X,Y, t) in Equation (21) as

Pi j (F�(X, t))= Pi j (F(X,Y, t))+��Pi j�Fmn

∣∣∣∣FF1mn+O(�2) (A2)



which after substituting (A2) into (A1) gives

Pi j (F�(X, t))= P0i j+�P1

i j+O(�2) (A3)

where

P0i j = Pi j (F(X,Y, t)) (A4)

P1i j =

�Pi j�Xk

∣∣∣∣XYk+ �Pi j

�Fmn

∣∣∣∣FF1mn (A5)

Inserting the asymptotic expansions (12) and (A3) into the equation of motion (1) and exploitingthe definition of the spatial derivative (17) yield

�−1�Pi j�Y j+ �P1

i j

�Y j+bXi (X,Y)−�X (X,Y)

�2u0i (X, t)

�t2+O(�)=0 (A6)

Substituting (A4) and (A5) into (A6) and identifying terms with equal powers of � yields thefollowing equilibrium equations:

O(�−1) : �Pi j�Y j=0 (A7)

O(�0) : �Pi j�X j+ �Gi j

�Y j+bXi (X,Y)−�X (X,Y)

�2u0i (X, t)

�t2=0 (A8)

where

�Gi j

�Y j= �Pi j

�Xk�Y j

∣∣∣∣XYk+ �Pi j

�Fmn�Y j

∣∣∣∣FF1mn+

�Pi j�Fmn

∣∣∣∣F

�F1mn

�Y j(A9)

Note that as Gi j is not a periodic function, integrating (A8) over the unit cell domain would notyield the equation of motion (29).

ACKNOWLEDGEMENTS

This work was supported by the Office of Naval Research grant number N000140310396 and the AirForce Research Laboratory grant number 478-05-506E.

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