Comput Mech DOI 10.1007/s00466-016-1354-3
ORIGINAL PAPER
Towards an efficient two-scale approach to model technical textiles Sebastian Fillep1 · Julia Mergheim1 · Paul Steinmann1
Received: 4 March 2016 / Accepted: 8 November 2016 © Springer-Verlag Berlin Heidelberg 2016
Abstract The paper proposes and investigates an efficient two-scale approach to describe the material behavior of technical textiles. On the macroscopic scale the considered textile materials are modeled as homogeneous by means of shell elements. The heterogeneous microstructure, which consists e.g. of woven fibers, is explicitly resolved in representative volume elements (RVE). A shell-specific homogenization scheme is applied to connect the macro and the micro scale. The simultaneous solution of the macroscopic and the nonlinear microscopic simulations, e.g. by means of the FE2 method, is very expensive. Therefore, a different approach is applied here: the macro constitutive response is computed in advance and tabulated for a certain RVE and for different loading scenarios. These homogenized stress and tangent values are then used in a macroscopic simulation without the need to explicitly resort to the microscopic simulations. The efficiency of the approach is analyzed by means of numerical examples. Keywords Computational homogenization · Technical textiles · Shell kinematics · Tangent calculation · Tabulated macro constitutive model
1 Introduction The mechanical behavior of textiles is influenced by the fiber material and the structural arrangement of the fibers at the micro scale. The structure of the fibers may be either periodic,
B 1
Sebastian Fillep
[email protected] University of Erlangen-Nuremberg, Egerlandstr. 5, 91058 Erlangen, Germany
e.g. woven or knitted, or randomly assembled like felt or nonwovens. To better understand the usually nonlinear behavior of technical textiles, modelling and simulation approaches on different length scales were introduced. The macroscopic behavior of textiles is e.g. simulated in [31,32], using special finite elements which account for the yarn structure of the textile. Numerical methods and novel element formulations addressing locking phenomena in textiles under large strains are developed in [35,36]. Comparisons of macroscopic simulations of a forming process with experimental observations are given in [39] for non-crimp fabrics and in [40] accounting for temperature influences. In [37] a lattice model is developed to model the yarn structure of textiles, which is then included in a quasicontinuum method to investigate the mechanical reliability of electronic textiles [38]. Further microscopic models and simulations can be found in [29,30], where the software package WiseTex is introduced, which models the elastic and damage behavior of different yarn structures of textiles and uses homogenization to predict macroscopic properties. The smallest length scale is considered in [33,34], where each fiber of a yarn of a textile is modeled by beam elements which interact by certain contact conditions. In the present contribution we assume that the yarns are small as compared to the macroscopic dimensions of the textile and that the textile can be considered homogeneous at the macroscopic scale. The heterogeneities, i.e. the yarns and contact zones, are explicitly resolved at the microscopic scale. The challenge is to obtain information of the material behavior from the micro-level and transfer this data to the macro scale. Different two-scale modeling and simulation approaches were developed to tackle this problem. Mainly two approaches of multiscale methods can be distinguished [1,2]: sequential and integrated ones. In sequential
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methods a macroscopic constitutive equation is developed by means of detailed, numerical investigations of mechanical tests of the microscopic sample. These results are processed to a constitutive law, which is provided together with material parameters for the macroscopic calculation as e.g. in [3] for elasto-visco-plastic materials. The advantage of the method is clearly that a so developed constitutive law can be reused for several calculations. Sequential approaches to model textiles were introduced in [5,6], where also anisotropic damage was taken into account. The accuracy of this method is restricted by the need to a priori choose an otherwise unknown macroscopic constitutive law that should capture the relevant microscopic effects. For more complex constitutive behavior these a priori defined laws do not provide satisfying solutions at the macro scale, as shown in [4]. Integrated schemes solve the microscopic and the macroscopic simulations simultaneously, as demonstrated in e.g. [8–10,15,16]. The constitutive behavior for each macroscopic point is obtained from a micro calculation. The microscopic level is modeled by means of an appropriate representative volume element (RVE). The application of suitable homogenization techniques permits to transfer information between the scales [7]. A straightforward method is the FE2 -scheme. The RVE is evaluated for each macroscopic integration point at each increment to obtain the macroscopic response. As highlighted in [26], this method is also applicable for large deformations, rotations and nonlinear material behavior. Contributions that deal with homogenization problems with contact are [41–43]. Both scales are connected by requiring the equality of the internal macro and micro power densities. For the macro to micro scale transition the macroscopic deformation is applied to the RVE. From the micro calculations the macroscopic stress resultants can be extracted [13,22]. The advantage of the FE2 -scheme is its accuracy [11], the drawback that it lacks efficiency. During a two-scale simulation the stress and the tangent, needed in a Newton scheme at the macro scale are computed for each macroscopic integration point at each deformation increment [21]. Thereby, neither the stress nor the tangent data are stored. If multiple integration points follow the same deformation path the same stress and tangent data is repeatedly computed. This problem becomes an important issue, when the computational costs for one RVE solution increase, as e.g. for textiles due to the solution of a contact-multibody problem at the fiber level. To avoid the a priori prediction of a phenomenological constitutive law as well as the inherent numerical inefficiency of the FE2 -method an alternative technique is elaborated here. A tabulated macro constitutive model is specified wherein the homogenized stresses and tangents of the microcalculations are stored and are then available for the macro calculation. Therewith, the numerical effort is reduced significantly. The tabulated macro constitutive model is computed
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offline and systematically for a prescribed deformation space in the sense of a numerical laboratory [17,18,28]. The deformation space is discretized, discrete deformations are applied to the RVE and the homogenized stresses and tangents are derived and stored. The homogenized tangents are computed numerically by a forward difference approximation, as suggested in [12]. Thus, the a priori definition of a macroscopic constitutive law is avoided but homogenized microscopic material data is transferred to the macroscopic scale in terms of the tabulated macro constitutive model. In this contribution the Lagrangian setting for finite hyperelasticity is chosen [14]. History-dependency of the material behavior and stability phenomena are neglected, although friction appears between the fibers at the micro-level, but its influence is assumed to be small. The tabulated macro constitutive model is here computed for woven textiles at the micro-level and presented in a format which is suitable for shell elements at the macro level [19,21]. A similar strategy could also be adopted for the homogenization of rope-like textiles to macroscopic beam elements [20]. The paper is organized as follows: The macroscopic problem is introduced in the next section. The shell kinematics are re-iterated and the shell-specific internal power density is elaborated. After the specification of the microscopic problem, a shell specific format of the Hill–Mandel condition is derived, which permits the coupling of the length scales. In Sect. 5, the computation of the shell specific tangent by a forward differentiation approximation is described. Together with the homogenized stress this tangent is stored in a tabulated macro constitutive model for later use on the macro level. Selected examples for the introduced methods i.e. computations of the tabulated macro constitutive model for selected deformations and a two scale simulation are given in Sect. 7.
2 Macroscopic problem At the macroscopic scale technical textiles can be efficiently modeled by means of shell elements. The shell formulation which will be recapitulated in this section accounts for large strains and rotations, for a non-zero stretch in thickness direction and is suitable for thin and thick shells. It is based on the shell kinematics elaborated in [24,25] and was used within a homogenization framework already in our previous contribution [19]. The shell formulation is repeated here for the sake of clarity. The shell kinematics will be described, followed by the definition of the internal power density which is later required to formulate the Hill–Mandel equation. 2.1 Shell kinematics A body B is a collection of material points P with X denoting the positions of P in the material configuration B0 at time
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G2
θ3 G3 A3
B0
θ2
G1 A2
A1
Bt
M0
θ1
θ3
g2
Mt
g3
g1
u
XM
E 3 = e3 X E 2 = e2
a2
a3
ϕ
θ2
θ1
a1 ϕM
E 1 = e1
Fig. 1 Kinematics of a shell in the material (left) and in the spatial configuration (right)
t0 in the three-dimensional Euclidean vector space E3 . The nonlinear deformation map to the spatial configuration Bt is defined as x = ϕ(X, t) ,
(1)
where x indicates the position of the material point P at time t > t0 . The geometry and deformation of a shell are described by curvilinear convective coordinates θ i , as given in Fig. 1. Subsequently, Latin indices range from 1 to 3 and Greek indices range from 1 to 2. The middle surface of the shell M is defined by θ 3 = 0. The material position and the nonlinear deformation map are then specified as X(θ i ) = X M (θ α ) + θ 3 D(θ α ) and α
α
ϕ(θ ) = ϕ M (θ ) + θ d(θ ) . i
3
(2)
The material and spatial vectors X M and ϕ M provide a parametric representation of the middle surface of the shell in the material and the spatial configuration, D and d denote the corresponding directors, respectively. The parameter θ 3 ∈ [− 21 H0 , 21 H0 ] determines the position of a point normal to the middle surface in the material configuration, whereby H0 is the material thickness of the shell. Hence, the possibly extensible director d captures changes in the cross sectional direction. The transverse stretch is defined as λ = ||d|| = Ht /H0 with || D|| = 1. The covariant basis vectors Aα and aα in the middle surface of the shell are computed from the partial derivative of the material placement X M and the spatial position ϕ M with respect to the curvilinear coordinates θ α Aα =
∂ XM ∂ϕ M = X M,α ,aα = = ϕ M,α . α ∂θ ∂θ α
(3)
Furthermore, the vectors A3 and a3 are introduced as normal to the middle surface M to span the shell space. In the
material and in the spatial configuration these vectors satisfy the orthogonality conditions Aα · A3 = 0 and aα · a3 = 0 .
(4)
Hence, in particular the transverse shear deformations are captured by the difference between d and a3 . The covariant basis of the shell space has to be extended to a threedimensional formulation. From Eq. (2) covariant basis vectors for the description of the shell body can be derived in analogy to Eq. (3) as ∂X ∂X = Aα + θ 3 D,α , G 3 = 3 = D , α ∂θ ∂θ ∂ϕ ∂ϕ 3 g α = α = aα + θ d ,α , g 3 = 3 = d . ∂θ ∂θ
Gα =
(5)
The corresponding contravariant basis vectors result from the relations G i · G j = δ ij
and
g i · g j = δ ij ,
(6)
with the Kronecker delta δ ij . The covariant basis vectors are tangents to the coordinate lines and the contravariant basis vectors are normal to the coordinate surfaces. Finally, the deformation gradient F is defined with the covariant basis vectors from Eq. (5) as F=
∂ϕ = g i ⊗ G i = aα + θ 3 d ,α ⊗ G α + d ⊗ G 3 . ∂X (7)
The variation of the deformation gradient F expressed in shell specific kinematic quantities thus reads δ F = δaα + θ 3 δd ,α ⊗ G α + δd ⊗ G 3 .
(8)
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As a strain measure the Green–Lagrange strain E is defined as
With the tractions t i = U INT =
1 T [F · F − I] = E i j G i ⊗ G j with 2 1 E i j = [gi j − G i j ] , 2 E=
G P · G i Eq. (13) renders
δa1 · t α dθ 3 θ 1θ 2
+ δd ,α ·
(9)
√
θ3
θ 3 t α dθ 3 +δd ·
θ3
where I is the identity tensor. The covariant metric coefficients G i j = G i · G j in the material configuration and gi j = g i · g j in the spatial configuration follow from the covariant basis vectors (5). As a result, the coefficients of the Green–Lagrange strain can be ordered by the power of the thickness coordinate θ 3
t 3 dθ 3 dθ 2 dθ 1 . (14)
θ3
Finally, the internal power density can be expressed as U INT = θ 1θ 2
[δa1 · nα + δd ,α · mα + δd · n3 ]dθ 2 dθ 1 , INT ℘M
(15) Ei j =
E i0j
+θ
3
E i1j
+ [θ ]
3 2
2 E αβ
.
(10)
1 = E 1 and the second order terms The first order terms E α3 3α 2 E αβ are in general neglected for thin shells, which are considered in this contribution.
with the stress resultants ni = t i dθ 3 , mα = θ 3 t α dθ 3 . θ3
(16)
θ3
2.2 Internal power density of a shell
Further the Piola–Kirchhoff stress S = F −1 · P is defined as
The internal virtual power U INT is defined as the volume integral of the internal virtual power density ℘ INT per unit volume in B0 and can be specified in work conjugate quantities e.g. the Piola stress P and the variation of the deformation gradient F as
S = S i j [G i ⊗ G j ] .
U
INT
=
℘
INT
dV =
B0
P : δ Fd V .
(11)
B0
For a shell, the volume integral is transferred to the parameter space of the curvilinear coordinates U
INT
=
√
θ 1θ 2θ 3 INT with ℘M :=
G P : δ Fdθ dθ dθ =
√
2
1
INT ℘M dθ 2 dθ 1 ,
θ 1θ 2
G℘ INT dθ 3 ,
(12)
where the volume element d√ V in the material configura√ tion B0 is defined as d V = Gdθ 1 dθ 2 dθ 3 with G = [G 1 × G 2 ] · G 3 . Introduction of (8) into (12) renders δaα · θ 1θ 2
+
√ θ3
θ 1θ 2
δd · θ 1θ 2
θ3
with N ∈ [0, 2] denoting the power of the thickness coordinate.
√ θ 3 G P · G α dθ 3 dθ 2 dθ 1
√ θ3
G P · G 3 dθ 3 dθ 2 dθ 1 .
3 The microscopic boundary value problem The heterogeneous microstructure of technical textiles, i.e. the particular fibers, are explicitly discretized at the micro scale. Exemplarily, an RVE for a woven textile is illustrated in Fig. 2 (right). The heterogeneous structure within the RVE is considered as a standard continuum. The microscopic defor¯ and the microscopic deformation ¯ X) mation map ϕ¯ = ϕ( gradient ¯ = ∂ ϕ¯ F ∂ X¯
G P · G α dθ 3 dθ 2 dθ 1
(19)
are defined. The equilibrium condition at the micro scale is given by the quasi static balance of momentum without body forces as
θ3
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δd ,α ·
+
ij
with the coefficients S N that are defined by pre-integration over the shell thickness √ ij S N = [θ 3 ] N S i j Gdθ 3 , (18)
3
θ3
U INT =
(17)
(13)
Div P¯ = 0 ,
(20)
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∂ B¯0,N
¯ θ¯ i ; θ α ) = X M (θ α ) + θ¯ i Ai (θ α ) . X(
(22)
The microscopic covariant basis vectors are
∂ B¯0,D
¯ θ¯ i ; θ α ) + w( ¯ θ¯ i ) , ¯ θ¯i ; θ α ) = F(θ α ; θ 3 = 0) · X( ϕ(
Fig. 2 Determination of the boundaries of the RVE
¯ The material behavior of with the microscopic Piola stress P. each fiber in the continuum body micro structure is described by a suitable constitutive law, e.g. in terms of the strain energy ¯ and thus the Piola stress is computed as ¯ = ( ¯ F) density
(21)
Contact interactions appear between the fibers. In this contribution friction is considered by the application of Coulomb’s law of friction. A detailed description can be found in [23]. The boundary ∂ B¯0 of the material filled part of the RVE can be split into three non-overlapping parts subjected to Neumann conditions on ∂ B¯0,N , to Dirichlet conditions on ∂ B¯0,D and unilateral contact conditions with sticking and Coulomb friction between the fibers on ∂ B¯0,C , see Fig. 2.
4 Homogenization for shells The RVE which describes the microstructure at a certain macroscopic position θ α is spanned by the macroscopic covariant basis vectors Ai of the middle surface in terms of the microscopic rectilinear coordinates θ¯i , as illustrated in Fig. 3. The microscopic coordinates of a point in the RVE are then given as Fig. 3 Macroscopic shell (left) and microscopic RVE (right): the RVE is spanned by the covariant basis vectors Ai of the macroscopic middle surface
(23)
whereby the coordinates in thickness direction are assumed to coincide for the micro and the macro scale θ 3 = θ¯ 3 . ¯ θ¯i ; θ α ) is given by an The microscopic deformation ϕ( ¯ affine part and a fluctuation w
∂ B¯0,C
¯ ¯ F) ∂ ( P¯ = . ¯ ∂F
¯i ¯ ¯ i = ∂ X = Ai and G ¯ i = ∂ θ = Ai , G ∂ X¯ ∂ θ¯i
A2
(24)
where F(θ α ; θ 3 = 0) is the macroscopic deformation gradient evaluated at the middle surface position X M (θ α ) and ¯ θ¯ i ) denotes the microscopic fluctuation field. By applyw( ing the gradient with respect to the material position ∇ X¯ the ¯ is given by microscopic deformation gradient F ¯ = F +∇¯w F X ¯ .
(25)
A basic concept of scale transition is the equality of the macroscopic and the averaged microscopic internal power density. For the shell specific homogenization a power averaging theorem is introduced, which is a special format of the known Hill–Mandel condition. The macroscopic internal power has to be equal to the microscopic internal power ¯ 0 of the RVE averaged over the middle surface M 1 ℘M = ¯ A0
℘¯ INT d V,
INT
(26)
B¯0
¯ 0 ) is the area measure of the middle where A¯ 0 := area(M surface in the parameter space. A characteristic of the shell specific homogenization scheme is that the microscopic RVE is not related to a point but to a straight line normal to the middle surface. This is a result of the pre-integration over the thickness direction of the RVE to assign all quantities to the middle surface. The microscopic internal power density ℘¯ INT is expressed with work conjugated quantities like the variation of the deformation gradient and the Piola stress
A1 A3
A1 A2
A3
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(27)
Replacing the internal power densities on both sides of the Hill–Mandel condition by the integrand of equation (15) or its microscopic counterpart, respectively, results in δaα · nα + δd ,α · mα + δd · n3 1 ¯ α + δ d¯ · n¯ 3 d θ¯ 2 d θ¯ 1 . δ a¯ α · n¯ α + δ d¯ ,α · m = A¯ 0
5 Tangent calculation
θ¯ 1 θ¯ 2
(28) The micro stress resultants are defined similarly as in equation (16) n¯ i :=
√
A P¯ · Ai dθ 3 =
θ¯3
√
¯ α := m
¯t i d θ¯ 3 ,
θ¯3
Aθ 3 P¯ · Aα d θ¯3 =
θ¯3
θ 3 ¯t d θ¯3 , α
(29)
θ¯3
√ with A = [ A1 × A2 ]· A3 . Since the micro deformation gradient consists of the macroscopic deformation gradient and the fluctuation gradient, two terms result if (25) is introduced into (28) and the Hill–Mandel condition reads eventually
δS = C : δE ,
with the fourth order tangent operator C. The tangent operator C can be determined numerically by a forward difference approximation, as suggested in [14] for phenomenological constitutive laws. The same approach is adopted here for the two scale modeling of shell structures. The homogenized tangent is derived from microscopic simulations of the RVEs for a prescribed macroscopic strain state E. Therewith, Eq. (33) can be rewritten and a linear increment of S follows as
S = C : E .
θ¯ 1 θ¯ 2
(30)
increments with a forward approximation. The index • Ei j indicates the particular coefficients of the Green–Lagrange strain that are perturbed, compare Eq. (10), whereby N ∈ [0, 2] denotes the power of the thickness coordinate. Firstly, the macroscopic stress state S is evaluated for a certain strain state E. This stress is then related to the stresses N S Ei j , which are computed for the perturbed strain states N
The following macroscopic quantities (or stress resultants) can thus be identified
mα :=
n¯ i d θ¯ 2 d θ¯ 1 ,
N
E Ei j = E + E Ei j to obtain the stress increments N
N
N
S Ei j ≈ S Ei j (E Ei j ) − S(E) .
(35)
The perturbation of the Green–Lagrange strain is defined as
θ¯ 1 θ¯ 2
1 A¯ 0
¯ α d θ¯ 2 d θ¯ 1 . m
N
(31)
θ¯ 1 θ¯ 2
By inserting Eq. (31) into (30) it follows that the part of the internal power related to the fluctuations averaged over the area of the RVE has to vanish . 1 0= A¯ 0
(34)
N
w w ¯ M,α¯ · n¯ α + δ d¯ ,α¯ · m ¯ α + δ d¯ · n¯ 3 d θ¯ 2 d θ¯ 1 . δw
1 ni := A¯ 0
(33)
N
θ¯ 1 θ¯ 2
1 A¯ 0
In the Lagrangian setting the relation between the variations of the macroscopic Piola–Kirchhoff stress δ S and the Green– Lagrange strain δ E is given by
The stress increment S Ei j is computed for particular strain
δaα · nα + δd ,α · mα + δd · n3 . 1 ¯ α + δd · n¯ 3 d θ¯ 2 d θ¯ 1 = δaα · n¯ α + δd ,α · m ¯ A0 +
If for example linear displacement boundary conditions are introduced, whereby the fluctuations are prescribed to be zero at the boundary of the RVE, the right hand side of (32) vanishes identically and thus the Hill–Mandel condition is satisfied, as was shown in detail in [19].
E Ei j := iNj ,
(36)
whereby iNj is a particular unit variation of the Green– Lagrange strain scaled by the perturbation parameter . The unit variation of the Green–Lagrange strain is defined as
1 = 1 − δi j [θ 3 ] N [E i ⊗ E j + E j ⊗ E i ] , 2
w w ¯ M,α¯ · n¯ α + δ d¯ ,α¯ · m ¯ α + δ d¯ · n¯ 3 d θ¯ 2 d θ¯ 1 . δw
iNj
θ¯1 θ¯2
with the unit vectors E i = E i in the material configuration. Inserting the stress increment (35) and the associated strain
(32)
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(37)
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increment (36) in the stress–strain relation (34) leads to the representation N
N
N
S Ei j = S Ei j (E Ei j ) − S(E) = C : iNj .
(38)
The evaluation of Eq. (38) for n φ independent unit variations of the Green–Lagrange strain results in the components of the tangent operator C at a certain deformation state. Remark For a better illustration of the tangent calculation method the Green–Lagrange strain tensor E of a thin shell is transferred to Voigt-like notation, indicated by •. This results in a vector with the relevant n φ = 5 components T 0 0 0 1 1 , E 22 , E 12 , E 11 , E 22 E = E 11 1 , E 2 , E 3 , E 4 , E 5 ]T , = [E
(39)
The first three entries represent the membrane strains and the first order components are the bending strains along the G 1 and G 2 direction, respectively. The thickness strains and the high order strains do not appear in a Kirchhoff shell element. Further, twist deformations are neglected, as the textile structure shows a negligible resistance for this deformation. In accordance, the Piola-Kirchhoff stress vector S is introduced as T S = S011 , S022 , S012 , S111 , S122 = [ S1, S2, S3, S4, S 5 ]T .
(40)
A transformation of Eq. (34) to Voigt-like notation renders the stress–strain relation
S= C · E,
(41)
with the tangent matrix C and the strain and stress increments
E and S. Following conceptually the former introduced scheme Eq. (38) becomes N
N
N
E E E iNj .
S ij = S i j ( E ij ) − S( E) ≈ C ·
(42)
For the sake of clarity the stress related to the k-th unit variaN
tion of the strain is renamed from • Ei j to •k with k ∈ [1, n φ ], where the representation of the unit variation of the strain is identified by the k-th entry of E. Therewith, Eq. (42) reads k k k k .
S = S ( E ) − S( E) ≈ C ·
(43)
After a rearrangement the k-th column of the tangent operator C is given by
T 1 k k 1 k k S (E ) − S ]T , S( E) = [ C =
(44)
1 2 n C ,..., C φ ]. The computational costs with C = [ C , for the calculation of one tangent operator at a prescribed deformation F are n φ + 1 RVE evaluations. Initially, the stress state S( E) for the current strain E has to be computed. Thereafter n φ evaluations of the boundary value problem of the RVE render the components of the tangent, i.e. one evaluation for each row of C.
6 Tabulated macro constitutive model The tabulated macro constitutive model is derived systematically for a certain section of the deformation space. Thereby the deformation space is spanned by the relevant independent coefficients of the deformation gradient. This results in a n λ -dimensional vector space, where n λ is the number of relevant deformation gradient coefficients. The position of a point in this deformation space is given by T . F λ = Fλ1 , Fλ2 , . . . , Fλn λ −1 , Fλn λ
(45)
The section of the deformation space is defined a priori with i i , Fλmax ]. It is then discretized by equidistant Fλi ∈ [Fλmin nodes in each direction, with the distance Fλi . Thus, the discretization results in Nλi nodes in each direction of the deformation space, where Nλi =
i i Fλmax − Fλmin
Fλi
+1.
(46)
The overall number of nodes is then given by Nn =
nλ
Nλi .
(47)
i=1
At these nodes of the deformation space RVE-simulations have to be performed to determine the macroscopic stresses and tangents, as described in Sect. 5. Between the nodes, n λ -dimensional finite element interpolations are introduced to approximate the macroscopic stress and tangent for intermediate values of F. If the resolution of the deformation space discretization is increased, i.e. if Fλi is reduced and the mesh is refined, the capability to capture non-linearities in the stress–strain relation is increased. In accordance with the accuracy, the numerical effort is also increased by smaller
Fλi , as the stress and the tangent have to be evaluated for more deformation nodes. Once the tabulated macro constitutive model is computed for a given macroscopic deformation state F λ it provides the
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macroscopic stress state S and the macroscopic tangent C, which can then be used within the Newton–Raphson solution scheme at the macro level. Remark
a macroscopic problem. The ratio of the CPU-times follows as n g n inc n it tch = . tmm Nn It is obvious that the tabulated macro constitutive model approach becomes more efficient if larger problems, i.e. with a higher number of Gauss points n g , are simulated for more load steps. Furthermore, once computed the tabulated macro constitutive model can be reused to simulate other macroscopic problems with the same underlying micro structure. To make the computation of the tabulated macro constitutive model more efficient symmetries may be exploited, which have to be estimated from the micro structural assembly. Further, to reduce the numerical effort instead of the deformation space the strain space is discretized. This reduces the dimension of the tabulated macro constitutive model space, e.g. in the current example from six to five.
To compare the computational costs of a FE2 -simulation and the tabulated macro constitutive model approach a simple estimation is presented. The macroscopic model is assumed to contain n g = 2000 integration points (e.g. 500 quadrilateral, linear shell elements) and n inc = 4 increments with n it = 3 iterations per increment that have to be performed for the Newton–Raphson solution method for one load step. Further, the size of the macroscopic shell tangent is of the dimension of n φ = 5, therefore six calculations of the microscopic boundary value problem have to be performed to compute the macroscopic tangent for one integration point per iteration. One micro calculation is considered to last tm = 1000 s. Applying the FE2 -method, the CPU-time needed for the overall, non-parallelized simulation for one load step is
7 Computational examples
tch = n g [n φ + 1]n inc n it tm = 1.44 · 108 s .
7.1 Setup
For this rough estimation CPU-times for the macro calculation itself and data transfer times are neglected, as they are small compared to the time needed for the RVE evaluation. For an application of the tabulated macro constitutive model, it is assumed that the size of the deformation space is known, e.g. from a macroscopic simulation using a similar phenomenological hyperelastic material law. Alternatively, the deformations space can be extended during the macro calculation by additional microscopic simulations. For the present example, the resulting n λ = 6 deformation directions T 0 0 0 0 1 1 , F22 , F12 , F21 , F11 , F22 F λ = F11 T = Fλ1 , Fλ2 , Fλ3 , Fλ4 , Fλ5 , Fλ6
(48)
have to be discretized. The deformation directions are discretized by Nλi = 4 nodes which is similar to the number of load increments n inc in the FE2 -simulation. The tangent calculation on the micro-level is still based on the former assumptions i.e. one tangent calculation requires six RVE simulations. The computation time for generating the tabulated macro constitutive model is then tmm = Nn [n φ + 1]tm = 2.46 · 107 s . This CPU-time is almost of the same magnitude as the one required for the FE2 -method. However, usually the computation of more than just five increments is required to solve
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For an illustration of the proposed method a square cutout of a plain woven textile, structured by fibers with lenticular cross sections, is chosen as the RVE, see Fig. 4. The RVE has a size of 10 × 10 × 2.6mm3 . The contact friction parameter is chosen to be constant μ f = 0.2. For all simulations isotropic hyper-elastic St.-Venant material behavior is assumed for the fibers ¯ + 2μ E ¯ , S¯ = λ tr E
(49)
where λ and μ are the Lamé constants. The parameters are assumed to be λ = 1150 MPa and μ = 770 MPa, which is characteristic for polyamide. The simulations are performed using the commercial finite element code MSC.Marc® . All problems discussed are solved within a Newton–Raphson strategy and all microscopic examples are discretized with 2200 20-noded hexahedral elements to capture properly the G3
G2
G1
Fig. 4 The RVE is composed of four woven prismatic fibers
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contact behavior. The differentiation parameter is chosen to be = 0.001. Nodes in contact are detected by a node-to-surface strategy whereby the contact in normal and tangential direction is captured by a penalty method.
the deformation space the evaluation of the stress by 5-by-5 (cross marker) nodes renders good results. For more nodes the stress coefficient shows no further increase in accuracy. 33 is captured, as illusFurther, the tangent coefficient C trated in Fig. 6a–c. The tangent calculations lead to almost constant values with superimposed irregular fluctuations. These fluctuations result from the nonlinearity of the contact, i.e. sticking and sliding of the fibres, together with the inaccuracies due to the numerical differentiation to compute the tangent. With larger deformations the fluctuations increase, as the interaction effects have more influence. Since the fluctuations are irregular and comparably small they cancel out if a certain deformation path is considered.
7.2 Computation of the tabulated macro constitutive model: shear To illustrate the introduced method different sections of the deformation space for the computation of the tabulated macro constitutive model for thin shells are investigated in detail. Thereby the position in the deformation space is given by the vector F λ . In the first example the section of the deformation space and thus the resulting tabulated macro constitutive model with non-zero shear deformations , Fλ3 , Fλ4 ∈ [0, 0.3] is considered. All other deformations are assumed to vanish, i.e. Fλ1 = Fλ2 = 1 and Fλ5 = Fλ6 = 0. The stress–strain relation is computed to be nearly linear, as illustrated in Fig. 5a–c for different resolutions, which is in good agreement with experimental results, e.g. in [27]. The interpolation between the exact values at the nodes or deformation space discretization is chosen to be bi-linear. For this section of
In the second example the biaxial tension section of the deformation space is chosen, i.e. Fλ1 , Fλ2 ∈ [1, 1.15], and is discretized with 13-by-13 equidistant nodes. At these nodes the stress and the tangent are evaluated by RVE calculations with the procedure described in Sect. 6. The stress coefficient 11 , i.e. the normal stiffness S 1 and the tangent coefficient C
0.3
0.3
0.1
15 10
0.05
25
0.15
20 0.1
S3
20
30
15
5 0
0
0.05
0.1
0.15
0.2
0.25
0.3
35 30 25
0.15
20 0.1
10
0.05
40
0.2
Fλ4 [ ]
25
35
[N/mm2 ]
0.15
45 0.25
40
0.2
Fλ4 [ ]
30
[N/mm2 ]
35
0.2
Fλ4 [ ]
0.25
40
S3
0.25
45
15 10
0.05
5
0
0
0
0.05
0.1
0.15
0.2
0.25
0.3
5
0
0
0
0.05
0.1
Fλ3 [ ]
Fλ3 [ ]
(a) 2-by-2 evaluation points.
[N/mm2 ]
45
S3
0.3
7.3 Computation of the tabulated macro constitutive model: biaxial tension
0.15
0.2
0.25
0.3
0
Fλ3 [ ]
(b) 5-by-5 evaluation points.
(c) 13-by-13 evaluation points.
Fig. 5 Stress coefficient S 3 for the shear section in the deformation space for the computation of the tabulated macro constitutive model discretized with bi-linear elements 80
0.3
85
79
0.1
78
75
0.15
74 73
0.1
72 71
0.05
0.05 77.5 0
0
0.05
0.1
0.15 Fλ3 [
0.2
0.25
0.3
]
(a) 2-by-2 evaluation points.
80 0.2 75
0.15 0.1
70
[N/mm2 ]
76
Fλ4 [ ]
77
[N/mm2 ]
78.5
0.25
78
0.2
Fλ4 [ ]
0.15
[N/mm2 ]
79
C 33
0.2
Fλ4 [ ]
0.25
79.5
C 33
0.25
C 33
0.3
80
0.3
0.05
70 0
0
0.05
0.1
0.15 Fλ3 [
0.2
0.25
0.3
]
(b) 5-by-5 evaluation points.
0
0
0.05
0.1
0.15 Fλ3 [
0.2
0.25
0.3
65
]
(c) 13-by-13 evaluation points.
33 for the shear section in the deformation space for the computation of the tabulated macro constitutive model Fig. 6 Shear stiffness coefficient C discretized with bi-linear elements
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Comput Mech 1.15
1.15
650
100
500 1.05
[N/mm2 ] C 11
40
1.05
550
Fλ2 [ ]
60
1.1
S1
Fλ2 [ ]
1.1
[N/mm2 ]
600 80
450
20
400 1
1
1.05
Fλ1
1.1
1.15
1
0
1
1.05
1.1
1.15
Fλ1 [ ]
[]
(a) 13-by-13 evaluation points.
(b) 13-by-13 evaluation points.
700
80
650
[N/mm2 ]
100
60 40
C 11
S 1 [N/mm2 ]
11 (right) for the biaxial tensile section in the deformation space for the Fig. 7 Normal stress coefficient S 1 (left) and tensile stiffness coefficient C computation of the tabulated macro constitutive model discretized with bi-linear elements with F 2λ = 0
20 0
1
1.05
1.1
Fλ1
1.15
[]
(a) Stress coefficient over deformation
600 550 500 450 400
1
1.05
Fλ1 [ ]
1.1
1.15
(b) Tensile stiffness coefficient over deformation.
(linear relation is depicted in gray for comparison).
11 for the biaxial tension section in the deformation space for the Fig. 8 Normal stress coefficient S 1 (left) and tensile stiffness coefficient C computation of the tabulated macro constitutive model discretized with bi-linear elements
in G 1 -direction are illustrated in Fig. 7. Between the nodes the discrete values are interpolated linearly. The stress coefficient S 1 increases progressively with increasing deformation in G 1 -direction but is nearly independent from transverse effects i.e. tension in G 2 -direction, see Fig. 7a. The tangent coefficient increases for small strains and converges to a constant value if Fλ1 is increased and Fλ2 is kept constant. If Fλ1 is constant the tangent remains almost constant if Fλ2 is increased, see Fig. 7b. To better illustrate this behavior, the stress coefficient S 1 and the normal stiff11 ness coefficient C are depicted in Fig. 8 for an increasing deformation Fλ1 and Fλ2 = 1. The degressive evolution of the tangent results in a slightly progressive stress strain relation. The changes in stiffness are a result of the modified contact zones during deformation. For Fλ1 ≤ 1.05 the sine-shaped fibers are straightened, the contact areas become larger and the stiffness increases continuously. For larger tensile strains, i.e. Fλ1 > 1.1, the stiffness remains nearly constant at about 650 N/mm2 and the stress increases linearly. At this deforma-
123
tion the fibers oriented in the tension direction are stretched to a flat shape and the contact zones do not change anymore. This microscopic behavior results in an almost constant 11 . The influence of the homogenized tangent coefficient C 2 transverse stretch Fλ on the stiffness is almost negligible. For small strains the stiffness increases slightly if Fλ2 is increased. This is also due to the growth of the contact zones. For large strains the contact zones are already fully developed and the influence of Fλ2 becomes insignificant. Once the contact areas do not change anymore, the macroscopic behavior is governed by the material behavior and the cross-section of the fibers. To estimate the resolution of the tabulated macro constitutive model that is necessary to approximate the material behavior of the RVE a study with a different number of nodes is performed, see Fig. 9 for the stress and Fig. 10 for the tangent. As expected, the evaluation of the deformation space section with only 2-by-2 nodes leads to an insufficient approximation of the nonlinear material behavior. A further
Comput Mech 1.15
1.15
40
1.05
20
1
1
1.05
Fλ1
1.1
1.15
60
40
1.05
20
0
1
[]
1
(a) 2-by-2 evaluation points.
1.05
Fλ1
[]
1.1
1.15
[N/mm2 ]
80
1.1
Fλ2 [ ]
[N/mm2 ]
60
100
S1
40
1.05
Fλ2 [ ]
Fλ2 [ ]
60
80
1.1
S1
80
1.1
100
[N/mm2 ]
100
S1
1.15
20
0
1
1
(b) 4-by-4 evaluation points.
1.05
Fλ1
[]
1.1
0
1.15
(c) 7-by-7 evaluation points.
Fig. 9 Normal stress coefficient S 1 for the biaxial tension section in the deformation space for the computation of the tabulated macro constitutive model discretized with bi-linear elements
500
1.05
450 1
1
1.05
1.1
600 1.1 550 500 1.05 450
450 1
1.15
1
1.05
1.1
1
1.15
400 1
1.05
(a) 2-by-2 evaluation points.
1.1
1.15
Fλ1 [ ]
Fλ1 [ ]
Fλ1 [ ]
[N/mm2 ]
650
Fλ2 [ ]
550
[N/mm2 ]
Fλ2 [ ]
500
1.05
[N/mm2 ]
Fλ2 [ ]
550
600 1.1
C 11
600 1.1
1.15
650
C 11
1.15
650
C 11
1.15
(b) 4-by-4 evaluation points.
(c) 7-by-7 evaluation points.
11 for the biaxial tension section in the deformation space for the computation of the tabulated macro Fig. 10 Tensile stiffness coefficient C constitutive model discretized with bi-linear elements 1.15
1.15
1.15
0.08
0.12
0.2
0.07
0.04
0.05
1
1
1.05
1.1
Fλ1
1.15
0
1
[]
1.05
1.1
1.15
0
0.04 0.03
1.05
0.02
0.02 1
0.05
[]
Fλ2 [ ]
0.06
0.06
1.1
ΔC 11
1.05
[]
1.05
0.08
ΔC 11
0.1
1.1
Fλ2 [ ]
0.15
[]
Fλ2 [ ]
1.1
ΔC 11
0.1
0.01 1
1
Fλ1 [ ]
1.05
Fλ1
1.1
1.15
0
[]
(a) Relative error to 2-by-2 evaluation
(b) Relative error to 4-by-4 evaluation
(c) Relative error to 7-by-7 evaluation
points.
points.
points.
11 for the biaxial tension section in the deformation space for the computation of Fig. 11 Relative differences of the tensile stiffness coefficient C the tabulated macro constitutive model discretized with bi-linear elements between the coarse discretized databases and the finer discretizations
increase of the number of evaluation nodes leads to a better approximation of the progressive stress and the strongly digressive tangent. To characterize the accuracy of the different resolutions of the material tangent in detail, the relative errors between the 13-by-13 tangent and the different tabulated macro constitutive model resolutions are illustrated in Fig. 11. As expected, the nonlinearity for small tensile
deformations in G 1 -direction is captured more properly with an increasing number of nodes. A dicretization with 7-by-7 nodes reduces the numerical effort to 29% compared to the the 13-by-13 tabulated macro constitutive model. The maximum relative error of 8% is acceptable while considering the savings in computational costs.
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Comput Mech
22 in G 2 Due to symmetry the stiffness coefficient C 1 direction shows the same behavior if Fλ and Fλ2 are interchanged. Further, a reduction of the numerical effort by an adaptive discretization of the deformation space is possible.
7.4 Path-dependencies As hyperelastic material is applied on the macro scale, this assumption has to be verified. The contact between the fibers may result in dissipative behavior. To capture this effect loading cycles are applied to the RVE. The hysteretic behavior for tension up to a maximum value of Fλ1 = 0.2, shear deformation up to Fλ3 = Fλ4 = 0.15 and bending deformation of Fλ5 = 0.05 is illustrated in Fig. 12. For the three deformations the hysteretic behavior and therewith the dissipation due to the contact is negligible. Therefore, the assumption of macroscopic hyperelasticity is considered to be valid for the chosen textile structure. This property has to be verified a priori for the selected micro structure and the section of the deformation space.
60 50 40 30 20 10 1
1.02
1.04
1.06
1.08
1.1
1.12
Fλ1 [ ]
(a) Tesion.
1.14
1.16
1.18
1.2
3
S1 S3
6 5
Stress resultants [N/mm2 ]
70
0
In the third example a macroscopic problem is solved by applying the tabulated macro constitutive model. Therefore, the strain space is discretized by four nodes in tension 1 ∈ [0.0, 0.05, 0.10, 0.15], see Fig. 14 direction lying at E (crossmarkers) and two nodes in the remaining strain direc3 , E 4 , E 5 ∈ [0.0, 0.05], i.e. Nn = 64. 2 , E tions E On the macro level a square shell is modeled, discretized by 64 linear 4-node thin-shell elements (MSC.Marc® element-number 139). The boundary conditions are applied according to Fig. 13 (left). A uniform prescribed displacement is applied on the right-hand side so that a constant deformation state is obtained. For the Newton–Raphson scheme both the macroscopic stress and the macroscopic tangent are obtained from the tabulated macro constitutive model. To evaluate the quality of the macroscopic simulation using the tabulated macro constitutive model, the RVE is loaded with the same deformation. The resulting stress
7
S1 S2
80
Stress resultants [N/mm2 ]
Stress resultants [N/mm2 ]
90
7.5 Two scale simulation using the tabulated macro constitutive model
4 3 2 1 0 -1
0
0.05
0.10
2.5 2 1.5 1 0.5 0 -0.5 0
0.15
S1 S4
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045 0.05
Fλ5 [ ]
Fλ3 [ ]
(b) Shear.
(c) Bending.
Fig. 12 Stress coefficient S 3 for the shear section in the deformation space for the computation of the tabulated macro constitutive model discretized with bi-linear elements [N/mm2 ]
[N/mm2 ] u
5.000e+02
9.24e+01
4.500e+02 4.000e+02
F
3.500e+02 3.000e+02 2.500e+02 2.000e+02 1.500e+02
G2
1.000e+02
G1
5.000e+01 0.000e+00
1 Fig. 13 Plot of the stress coefficient SMAC on the deformed macro shell with indicated boundary conditions (left) and a plot of the stress coefficient 1 ¯S on the deformed micro structure (right)
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Comput Mech
0
0
0.05
0.1
0.15
-2
0.1
8
0.05
6
0
4
-0.05
2 SMAC 2 SRVE
2
-0.1
2 SDIFF
0
0
0.05
0.1
0.15
Stress difference [N/mm2 ]
1 SDIFF
50
Stress [N/mm2 ]
Stress [N/mm2 ]
1 SRVE
0
10
2 1 SMAC
Stress difference [N/mm2 ]
100
-0.15
E1 [ ]
E1 [ ]
Fig. 14 Comparison of the stresses of a uniform tension at the macro scale evaluated with the tabulated macro constitutive model (black) and a 1 direct calculation with a RVE (grey dashed) and difference of the stress–strain relations SDIFF (red) Stamp
∅ 600
R
0 11
Blank holder Textile
R ∅ 320
20
∅ 1000
Shaping die
Fig. 15 Geometric set-up of the draping test for the validation of the methods. All measures in mm
S¯ is depicted in Fig. 13 (right). A significantly higher state stress level is reached in G 1 -direction locally than is captured by the homogenization procedure. Due to the homogeneous macroscopic strain field, the homogenized stress–strain relation of one RVE is equal to the macroscopic stress–strain solution obtained for the fully resolved structure. Therefore, the stress–strain relation of the macroscopic shell evaluated by the calculated tabu1 can be compared to lated macro constitutive model SMAC 1 . the homogenized stress–strain relation of one RVE SRVE 1 This is done for S , see Fig. 14 (left). The results are in very good agreement. The red curve indicates that the 1 < 0.05 with a maximum value largest difference is at E 1 1 1 SMAC − SRVE = −1.3 N/mm2 , where also the
SDIFF = tangent shows the highest nonlinearity. Further the difference is equal zero at the evaluation points (cross markers). Also S 2 , which is a result from transverse contraction effect, 1
is captured. Here the results also show very good agreement, see Fig. 14 (right). The red difference curve of the two results is also zero at the evaluation points with a maximal differ2 1 < 0.05 of SDIFF = −0.11 N/mm2 but the ence at E shape is not as smooth as in the former example. As this stress coefficient results from the compression of the fibers due to tension, the nonlinear contact interactions have a more significant influence to that stress state.
7.6 Validation: draping test of a textile For the validation of the introduced methods for the twoscale consideration of technical textiles the woven textile is loaded inhomogeneously. A draping test is performed with the experimental setup depicted in Fig. 15. A circular shaped region of the textile is loaded by a spherical stamp in negative
123
Comput Mech 90 120
400
0 tex Crr
60
200
G2 180
1.332e+01
0 iso Crr
300 150
[N/mm2 ] 1.213e+01 1.094e+01
30
G2
9.752e+00
100
8.564e+00 0
G1
G1
7.377e+00
ϑ[◦ ]
6.189e+00 5.001e+00 3.813e+00 2.626e+00
330
210
1.438e+00 240
300 270
0 (a) Comparison of the radial tensile stiffness Crr along the rotation angle ϑ starting from G1 = 0◦ for the woven textile (cyan) and an isotropic shell (magenta) for F = I.
(b) Von Mises stress state of the textile at a stamp path of z = 100 mm.
Fig. 16 Microscopic homogenized stiffness (a) and resulting macroscopic stress state (b) [mm] 2.629e+00 1.855e-01 -2.258e+00 -4.702e+00
G2
-7.145e+00 -9.589e+00
G1 G1
G2
-1.203e+01 -1.448e+01 -1.692e+01 -1.936e+01 -2.181e+01
(a) Radial displacement urr of the textile at a traverse path of the stamp of z = 100 mm.
(b) Deformed textile after the draping process [40].
Fig. 17 Comparison of simulation (a) and real textile (b)
G 3 -direction. The boundary of the textile is fixed by a blank holder. The geometry data can be found in Fig. 15. Stamp, blank holder and shaping die are modeled as rigid bodies. The textile is discretized by 5600 linear, thin shell elements. The contact parameters between the contact partners are chosen as μH = μG = 0.2. For these more complex loading conditions the tabulated macro constitutive model is extended and evaluated 2 ∈ [0.0, 0.05, 0.1, 0.15] for tension, E 3 ∈ [0.0, 1 , E at E 4 5 0.05, 0.1, 0.15, 0.20] for shear and E , E ∈ [0.0, 0.05] for bending deformation. This results in 320 evaluation points and therewith 1920 RVE-calculations. In contrast, for this example a direct coupling scheme takes 33600 RVEcalculations in each iteration.
123
In this example, the tension stiffness is mainly responsible for the magnitude of the stamp force whereas the relation of shear and bending stiffness influences wrinkling during the draping process [31]. It is expected that the woven textile shows transversal isotropic constitutive behavior, i.e. the material behavior is invariant with respect to rotations of 90◦ around the G 3 - axis. Along the G 1 and G 2 -axis i.e. in the direction of the woven fibers the stiffness of the textile is maximum, see Fig. 16a, for an angle of 45◦ the stiffness is minimum. Due to the anisotropy of the material a non-symmetric stress state occurs during the deformation. For a traverse path of the stamp of z = 100 mm the von Mises stress state, depicted in Fig. 16a, appears. Considering the radial displace-
Comput Mech 2500
[N/mm2 ] 1.188e+01
2000
1.067e+01 9.467e+00
G2
7.057e+00
G1
5.851e+00 4.646e+00
f [N]
8.262e+00
1500 1000
3.441e+00 2.236e+00
500
1.031e+00 -1.744e-01
0 0
(a) Von Mises stress state.
20
40
60
80
z [mm]
100
120
(b) Force-distance characteristic of the stamp.
Fig. 18 Stress state on the deformed textile at a stamp path of 110 mm (a). Stamp force versus stamp path. The initiation of instabilities can be observed by means of the decrease in the stamp force (b)
ments in Fig. 17a, larger displacements can be observed along the fiber orientations, i.e. in the G 1 and G 2 directions, than along the diagonal. Further the transversal isotropy is clearly considerable in the middle region of the textile. The displacement coincides qualitatively with the results from [31], see Fig. 17b. The stamp-force along the traverse path of the stamp is considered, see Fig. 18b, as this quantity is a characteristic measure for experiment. The resulting highly progressive curve is caused by the underlying nonlinear textile material and the loading, where initially bending and later tension appear as characteristic loading conditions. Up to a stamp path of z = 105 mm the stamp force is increasing. Thereafter a significant decrease of the stamp-force can be observed, see Fig. 18b. At this point localization effects occur below the blank holder, where shear bands appear. Within the shaping die wrinkling phenomena are caused by stability effects. The localizatoin zones predicted by the introduced method
G1
G2
zone of localization
Fig. 19 Draping test of a textile around a sphere. Initiation of the localization zone at the blank holder in G 1 and G 2 -direction; Figure from [40]
are similar to the ones experimentally observed in [40] for a woven textile, compare Fig. 19.
8 Conclusions An efficient two-scale approach to describe the material behavior of technical textiles was proposed. Therein textiles are modeled by shell elements on the macroscopic scales, which are related to volumetric RVEs on the micro scale which explicitly resolve the fibers and their contact zones. A computational homogenization scheme was introduced, which is based on a shell specific Hill–Mandel condition and relates homogenized forces and moments to certain macroscopic deformations. To reduce the computational costs of a direct two-scale simulation, as for example by means of the FE2 -method, the homogenized stresses and tangents of the RVE are precomputed numerically for a certain discretization of the deformation space. The stress and tangent values are stored in a tabulated macro constitutive model and can then be used for macroscopic simulations without requiring further microscopic simulations. The a priori definition of a macroscopic constitutive law is also avoided. The approach is restricted to hyperelastic materials. Its application to dissipative material behavior is only possible for special cases and would involve the computation of disctinctly more data points to capture the path dependency. The determination of the tabulated macro constitutive model was illustrated for certain deformations, discussing the required resolution of the deformation space. A comparison between a direct fully resolved simulation and the tabulated macro constitutive model approach showed that similar results are obtained whereby the material tangent approach is more efficient.
123
Comput Mech Acknowledgements This project was funded by the DFG (STE 544/40-2).
References 1. Feyel F (2003) A multilevel finite element method (FE2 ) to describe the response of highly non-linear structures using generalized continua. Comput Methods Appl Mech Eng 192:3233–3244 2. Feyel F, Chaboche J-L (2000) FE2 multiscale approach for modelling the elasto visco plastic behaviour of long fibre SiC/Ti composite materials. Comput Methods Appl Mech Eng 183:309– 330 3. van der Sluis O, Schreurs PJG, Brekelmans WAM, Meijer HEH (2001) Homogenisation of structured elastoviscoplastic solids at finite strains. Mech Mater 33:499–522 4. McHugh PE, Asaro RJ, Shih CF (1993) Computational modeling of metal matrix composite materials - III. Comparisons with phenomenological models. Acta Metall Mater 41:1489–1499 5. Peng X, Cao J (2002) A dual homogenization and finite element approach for material characterization of textile composites. Composites B 33:45–56 6. Takano N, Uetsuji Y, Kashiwagi Y, Zako M (1999) Hierarchical modelling of textile composite materials and structures by the homogenization method. Modell Simul Mater Sci Eng 7:207–231 7. Miehe C (2003) Computational micro-to-macro transitions for discretized micro-structures of heterogeneous materials at finite strains based on the minimization of averaged incremental energy. Comput Methods Appl Mech 192:559–591 8. Suquet P (1985) Local and global aspects in the mathematical theory of plasticity. In: Sawczuk A, Bianchi G (eds) Plasticity today: modelling, methods and applications. Elsevier, London 9. Ghosh S, Lee K, Moorthy S (1995) Multiple scale analysis of heterogeneous elastic structures using homogenization theory and voronoi cell finite element method. Int J Solids Struct 32:27–62 10. Ghosh S, Lee K, Moorthy S (1996) Two scale analysis of heterogeneous elastic-plastic materials with asymptotic homogenization and Voronoi cell finite element model. Comput Methods Appl Mech Eng 132:63–116 11. Geers MGD, Kouznetsova VG, Brekelmans WAM (2010) Multiscale computational homogenization: trends and challenges. J Comput Appl Math 234:2175–2182 12. Miehe C (1999) Computational micro-macro transitions and overall moduli in the analysis of polycrystals at large strains. Comput Mater Sci 16:372–382 13. Miehe C, Koch A (2002) Computational micro-to-macro transitions of discretized microstructures undergoing small strains. Arch Appl Mech 72:300–317 14. Miehe C (1996) Numerical computation of algorithmic (consistent) tangent moduli in large-strain computational inelasticity. Comput Methods Appl Mech 134:223–240 15. Kouznetsova VG, Geers MGD, Brekelmans WAM (2004) Multiscale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy. Comput Methods Appl Mech 193:5525–5550 16. Castañeda PP, Tiberio E (2000) A second-order homogenization method in finite elasticity and applications to black-filled elastomers. J Mech Phys Solids 48:1389–1411 17. Temizer I, Wriggers P (2006) An adaptive method for homogenization in orthotropic nonlinear elasticity. Comput Methods Appl Mech 196:3409–3423 18. Temizer I, Zohdi TI (2007) A numerical method for homogenization in non-linear elasticity. Comput Mech 40:281–298 19. Fillep S, Mergheim J, Steinmann P (2013) Computational modelling and homogenization of technical textiles. Eng Struct 50:68– 73
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20. Fillep S, Mergheim J, Steinmann P (2015) Computational homogenization of rope-like technical textiles. Comput Mech 55:577–590 21. Geers MGD, Coenen E, Kouznetsova VG (2007) Multi-scale computational homogenization of structured thin sheets. Model Simul Mater Sci 15:373–404 22. Kouznetsova VG, Brekelmans WAM, Baaijens FPT (2001) An approach to micro-macro modeling of heterogeneous materials. Comput Mech 27:37–48 23. Laursen TA, Simo JC (1993) A continuum-based finite element formulation for the implicit solution of multibody, large deformation frictional contact problems. Int J Numer Methods Eng 36:3451– 3485 24. Green AE, Zerna W (1954) Theoretical elasticity. Clarendon Press, Oxford 25. Betsch P, Gruttmann F, Stein E (1996) A 4-node finite shell element for the implementation of general hyperelastic 3D-elasticity at finite strains. Comput Methods Appl Mech Eng 130:57–79 26. Smit RJM, Brekelmans WAM, Meijer HEH (1998) Prediction of the mechanical behaviour of non-linear heterogeneous systems by multi-level finite element modeling. Comput Methods Appl Mech Eng 155:181–192 27. Liu L, Chen J, Li X, Sherwood J (2005) Two-dimensional macromechanics shear models of woven fabrics. Composites Part A 36:105–114 28. Yvonnet J, Monteiro E, He Q-C (2013) Computational homogenization method and reduced database model for hyperelastic heterogeneous structures. Int J Multiscale Comput Eng 3: 201–225 29. Verpoest I, Lomov SV (2005) Virtual textile composites software WiseTex: integration with micro-mechanical, permeability and structural analysis. Compos Sci Technol 65:2563–2574 30. Lomov SV, Ivanov DS, Verpoest I, Zako M, Kurashiki T, Nakai H, Hirosawa S (2007) Meso-FE modelling of textile composites: road map, data flow and algorithms. Compos Sci Technol 67: 1870–1891 31. Hamila N, Boisse P, Chatel S (2008) Finite element simulation of composite reinforcement draping using a three node semi discrete triangle. Int J Mater Form 1:867–870 32. Hamila N, Boisse P, Sabourin F, Brunet M (2009) A semi-discrete shell finite element for textile composite reinforcement forming simulation. Int J Numer Methods Eng 79:1443–1466 33. Durville D (2010) Simulation of the mechanical behaviour of woven fabrics at the scale of fibers. Int J Mater Form 3:1241–1252 34. Durville D (2009) A finite element approach of the behaviour of woven materials at microscopic scale. In: Mechanics of microstructured solids. Springer, Berlin, pp 39-46 35. Ten Thije RHW, Akkerman R, Huétink J (2007) Large deformation simulation of anisotropic material using an updated Lagrangian finite element method. Comput Methods Appl Mech Eng 196:3141–3150 36. Ten Thije RHW, Akkerman R (2008) Solutions to intra-ply shear locking in finite element analyses of fibre reinforced materials. Composites Part A 39:1167–1176 37. Beex LAA, Verberne CW, Peerlings RHJ (2013) Experimental identification of a lattice model for woven fabrics: application to electronic textile. Composites Part A 48:82–92 38. Beex LAA, Peerlings RHJ, Van Os K, Geers MDG (2015) The mechanical reliability of an electronic textile investigated using the virtual-power-based quasicontinuum method. Mech Mater 80:52– 66 39. Yu W-R, Harrison P, Long A (2005) Finite element forming simulation for non-crimp fabrics using a non-orthogonal constitutive equation. Composites Part A 36:1079–1093 40. Lin H, Wang J, Long A, Clifford M, Harrison P (2007) Predictive modelling for optimization of textile composite forming. Compos Sci Technol 67:3242–3252
Comput Mech 41. Tworzydlo WW, Cecot W, Oden JT, Yew CH (1998) Computational micro-and macroscopic models of contact and friction: formulation, approach and applications. Wear 220:113–140 42. Bandeira AA, Wriggers P, de Mattos Pimenta P (2004) Numerical derivation of contact mechanics interface laws using a finite element approach for large 3D deformation. Int J Numer Methods Eng 59:173–195
43. Hager C, Hueber S, Wohlmuth BI (2008) A stable energyconserving approach for frictional contact problems based on quadrature formulas. Int J Numer Methods Eng 73:205–225
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