Specifying Constituent Properties for a Composite Material
In multiscale material modeling, each composite material consists of one matrix material and one
or more inclusions including voids. In Abaqus/Explicit the composite material must consist of one matrix material and only one inclusion; the
inclusion material must be linear elastic with constant properties, and the matrix material
can be only linear elastic or elastic-plastic with isotropic hardening.
Mean-Field Homogenization for Linear Elastic Composites
The mean-field strain and stress in each phase I is
defined as
where
The averaged macro fields can be written as
where
is the volume fraction of the matrix phase and
is the volume fraction of the inclusion phase. For the single-inclusion case,
if all the constituents are linear elastic, the strain in the inclusion is
related to the strain in the matrix through a concentration tensor
.
The form of the concentration tensor defines the homogenization methods
below.
Voigt and Reuss Models
The simplest mean-field homogenization formulations are the Voigt and Reuss
models. These models do not take into account the shape or the orientation of
the inclusions; however, they provide upper and lower bounds of the macro
stiffness modulus and, therefore, can be used for validation. The Voigt model
is also used in a two-step approach to model composites with multiple
inclusions.
The Voigt model assumes uniform strain in the composite
,
which gives
The Reuss model assumes uniform stress in the composite
,
which gives
where
is the stiffness of the matrix and
is the stiffness of the inclusion.
Eshelby-Based Models
The more sophisticated mean-field homogenizations are based on Eshelby's
solution. In his 1957 paper (Eshelby, 1957), Eshelby solved a
single-inclusion problem described in
Figure 1.
Eshelby found that the strain inside the constrained ellipsoid is uniform
and given by
where
represents a point inside the inclusion and
is the eigenstrain. The Eshelby tensor (E) is a function
of the stiffness of the matrix and also the shape and orientation of the
inclusion. To calculate Eshelby's tensor usually involves numerical integration
over the surface of the ellipsoidal inclusion; however, analytical solutions
are available (Mura, 1987) for simple shapes such as a
sphere, cylinder, penny, oblate, and prolate. Based on Eshelby's solution, the
following models are proposed based on different assumptions about the
properties of the outer material surrounding the inclusions.
Mori-Tanaka Model
The Mori-Tanaka formulation assumes that each inclusion behaves like an
isolated inclusion and the strain in the matrix is considered as the far-field
strain. Therefore, the concentration tensor can be written as
Inverse Mori-Tanaka Model
The inverse Mori-Tanaka formulation assumes a high volume fraction of the
inclusion and permutes the properties of the matrix and inclusion in a single
inclusion problem, which gives
Balanced Model
The Mori-Tanaka and inverse Mori-Tanaka formulations give the upper and lower bounds of the
composite stiffness for low and high volume fractions. The balanced formulation is the
interpolative formulation, which can be written as
where
is the smooth interpolation function proposed by Lielens (1999)
User-Defined Concentration Tensor
You can define the concentration tensor directly by specifying all of its components. The user-defined
concentration tensor must be defined in the local coordinate system of the inclusion. You
can use homogenization models other than those listed above, calibration from experimental
results, or FE-RVE models to obtain the value of the
concentration tensor.
Incremental Mean-Field Homogenization for Nonlinear Composites
An incremental linearization approach is used to extend the homogenization
models to composites with nonlinear behavior. The tangent stiffness matrix is
used in place of the linear elastic modulus to compute the concentration tensor
Since the tangent matrix is not constant in the nonlinear case and is
usually a function of the strain, iterations are performed to guarantee a
converged solution of the strain increment in each constituent.
Isotropization Methods
Analytical expression of Eshelby's tensor is available only if the
inclusions are ellipsoidal and the matrix material is isotropic or transversely
isotropic. For matrix materials that are anisotropic, using the stiffness
directly to compute the Eshelby's tensor can result in stiffer prediction for
the composite behavior (Doghri, 2003). The isotropic part of the
tangent stiffness matrix can generally be written as
where
is the bulk modulus and
is the shear modulus.
and
are the volumetric and deviatoric part of the fourth-order identity tensor. The
method used to extract the isotropic part of the stiffness matrix is not
unique, and the accuracy of the prediction varies depending on the properties
of the matrix material.
General Method
The general method is the most common isotropization method, and it
extracts the isotropic part using
The general method can be used for any anisotropic tangent matrix. This
method is the default isotropization method.
Spectral Method
The spectral isotropization method can be used if isotropic hardening is
defined in the matrix material (Doghri, 2003). The isotropic part of the
tangent matrix is given by
where
is the elastic bulk modulus, is the elastic shear modulus,
is the yield stress, and
is the equivalent plastic strain.
Modified Spectral Method
For matrix materials with significant softening after yielding, both the
general method and the spectral method might give predictions that are too
stiff. The modified spectral method proposed by Selmi et. al. (Selmi, 2011) improves the accuracy of the
spectral method by evaluating the derivative of the yield stress at a larger
equivalent plastic strain
where , , and are the isotropization parameters. You can calibrate these parameters
by comparing the mean-field solution to experimental data or to FE-RVE solutions. The
isotropization parameters are irrelevant if the constituent does not have plastic behavior.
Isotropic Projection
Options are available to use the isotropic projection of the tangent matrix to compute different
parts of the concentration tensor. For better predictions, Doghri et al. (Doghri, 2010) recommended the following:
If the general method is used, only Eshelby's tensor is computed with the isotropic
projection.
If the spectral/modified spectral method is used, Hill's tensor given by is computed with the isotropic projection.
It is possible that some choices can lead to unphysical prediction of microlevel strain
resulting in unstable material behavior in the composite. In addition, in Abaqus/Standard analyses, the tangent stiffness matrix for the composite might lose its symmetry after
homogenization; therefore, you must use the unsymmetric solver to achieve convergence.
Multistep Homogenization
For composites with multiple inclusions, a multistep homogenization approach is used, as shown in
Figure 2. The composite is decomposed into "grains," with each grain
containing one inclusion family and the matrix. The inclusions in each family have the same
material properties, aspect ratio, and orientation. In the first step homogenization is
performed in each grain using the user-specified formulation; in the second step the Voigt
formulation is used to compute the properties of the overall composite.
An alternative approach is to use the Mori-Tanaka scheme in both the first and the second
step, assuming the average strain in the matrix is uniform across all grains. This approach
is equivalent to the direct Mori-Tanaka approach proposed by Benveniste (Benveniste et al., 1991). In an Abaqus/Standard analysis, the second approach has the drawback that it might result in an unsymmetric
effective modulus when the inclusions are misaligned and nonisotropic. In such cases, use of
the unsymmetric solver might be necessary.
You can specify the microstructure of the composite by giving the shape,
volume fraction, aspect ratio, and fiber orientation of each constituent. You
can use a distribution to define the volume fraction, aspect ratio, and the
orientation tensor (see
Distribution Definition).
The orientation tensor can be used only with three-dimensional solid elements.
Specifying the Shape, Volume Fraction, and Aspect Ratio
The volume fraction and aspect ratio can be specified as spatial
distributions. The aspect ratio is given by
as shown in
Figure 3.
The aspect ratio of a spherical inclusion is 1.0, and the aspect ratio of a
cylindrical inclusion is infinity; therefore, you do not need to specify the
aspect ratio in these cases.
Specifying a Fixed Orientation for the Inclusion and Void Phases
By default, the orientation of the inclusion or void phase is fixed and is
aligned with the x-axis of the local coordinate system
specified in the section definition. You can specify a different orientation by
giving the three components of the orientation vector in the local coordinate
system, as shown in
Figure 4.
The components of the orientation unit vector, ,
are
The local 3-direction of the inclusion (void included) is the projection of
the local z-axis defined in the section onto the surface
normal to the inclusion orientation (1-direction). If the local
z-axis is within 0.1 degree of being parallel to the
1-direction, the local 3-direction is the projection of the local
y-axis onto the surface. The local 2-direction is then at
right angles to the local 3-direction, so that the local 2-direction, local
3-direction, and the inclusion orientation (1-direction) form a right-handed
set. The local 2 and 3 directions are relevant only if the shape of the
inclusion (void included) is an elliptical cylinder.
Specifying a Second-Order Orientation Tensor in Abaqus/Standard
For composites containing short fibers, a second-order orientation tensor, , is usually used to describe the dispersion of the fiber orientation
where is the integral over the surface of the unit sphere (all possible
directions of ) and is the orientation distribution function
(ODF). The spatial distribution of the orientation
tensor is usually given by injection molding software and can be used directly to compute
the effective moduli of the composite material that is linear elastic. If the material is
nonlinear, the average fields given by must be computed during the analysis and the
ODF is required to calculate the integral. When only
the second-order orientation is available, Abaqus recovers the ODF using the approach proposed by Onat
and Leckie (1988)
for three-dimensional orientations, where and are the tensor basis functions
and and are the deviatoric versions of the orientation tensors
The fourth-order orientation tensor is required during the ODF reconstruction. It is
computed using a closure approximation, which is a formula that approximates the
fourth-order tensor in terms of the known second-order tensor. A popular closure
approximation is the hybrid closure approximation proposed by Advani and Tucker (1987), which is constructed using the interpolation
of linear and quadratic closure approximation
where the linear closure is given by
and the quadratic closure is given by
The weight function is defined as for three-dimensional orientations.
Only one constituent in the composite is allowed to have orientations specified with the
second-order orientation tensor.
Taking into account the symmetry of the ODF , we compute the integral numerically as follows:
where and . The numerical integral is computed by subdividing the macro point into subdomains, with each subdomain having the same fiber orientation . Then a multistep homogenization (as described in Figure 2) is used to compute the macrolevel solutions. The
ODF is more accurately recovered if a large value of
N is used; however, the computational cost and memory usage also
increases substantially.
When the fiber direction is random in the three-dimensional space, the second-order
orientation tensor is , and the ODF is constant across all
subdomains.
When the direct Mori-Tanaka homogenization approach is used, the fiber response can be
calculated either individually in each subdomain or just once using the averaged state
over all subdomains. In both cases only the averaged fiber state is available for output.
The maximum principal direction of the second-order orientation tensor is used as the
1-direction of the local coordinate system for the inclusion.
Specifying a Second-Order Orientation Tensor in Abaqus/Explicit
In Abaqus/Explicit, when a second-order orientation tensor, , is used to specify the dispersion of the fiber material, the direct
Mori-Tanaka approach is used. Because the fiber material can be only linear elastic with
constant moduli, homogenization and strain partitioning can be simplified and no numerical
integration is necessary. The following equation is used to compute the orientation
averaging terms, :
where , , , , and are the five constants of the tensor that is transversely isotropic. ODF
reconstruction is not necessary and is not performed.
The second-order orientation tensor is specified in the orientation definition of the
section where the multiscale material is assigned.
Composites with Thermal Expansion
Composites with thermal expansion are supported only in Abaqus/Standard.
A three-step approach is used to compute the macro response of a
thermoelastic composite (Pierard, 2004). The total macro strain is
applied to the composite in the first step with no temperature change. The
strain in each constituent is computed with the concentration tensor, and the
stress is updated accordingly. In Step 2 an equal temperature change is applied
to all constituents; by assuming uniform stress increment in all constituents,
the strain increment in the second step is computed in each constituent. In
Step 3 an equal and opposite strain increment computed in Step 2 is applied to
all constituents to recover the total macro strain; there is no temperature
change in this step. The total strain and stress in each constituent is
computed by superposing the solutions from all three steps. This approach is
also applied to the incremental formulation of nonlinear composites with
thermal expansion.
Thermal expansion is supported only for three-dimensional solid elements.
Composites with Viscoelasticity
Composites with viscoelasticity are supported only in Abaqus/Standard.
When one or more constituents have viscoelasticity defined in the time domain, the homogenization
approach described above is no longer appropriate because it does not take into account the
relaxation of stress with time. However, similar homogenization can be performed in the
frequency domain, and homogenized properties of the composite in the time domain can be
computed using homogenized properties at different frequencies. This homogenization approach
is applicable only when the viscoelasticity in the constituents is defined in the time
domain using a Prony series expansion and includes the following steps:
The shear and bulk moduli of all the constituents are transformed to the
frequency domain using Fourier transforms (see details in
Prony Series Parameters).
The mean-field homogenization is performed in the frequency domain the
same way as discussed above, with the exception that the moduli are complex
numbers instead of real numbers. The real part of the composite modulus is the
storage modulus, ,
and the imaginary part is the loss modulus, .
Assuming the time response of the composite can be written in a similar
form as that in the constituents where viscoelasticity is defined using a Prony
series expansionwhere
and
is the instantaneous elastic modulus of the composite, the homogenized storage
modulus of the composite, ,
can be written in the frequency domain as follows:
The parameters
are selected to be equally spaced numbers on a logarithmic scale in the range
.
and
are the minimum and maximum frequencies used to define the viscoelasticity in
all constituents. The parameters
are computed through a least-squares curve fit at frequencies
.
The thermorheologically simple temperature effects are not supported.
Other limitations include:
The inverse Mori-Tanaka and balanced homogenization formulations are not supported
when you use the orientation tensor to define the direction of the inclusion.
When you use the orientation tensor to define the direction of the inclusion, you can
define only transversely isotropic elasticity in the inclusion material, and no other
inclusions can be defined in the composite.
You cannot use the orientation tensor to define the direction of the inclusion in
which viscoelasticity is defined.
Distributions other than distributions of the orientation tensor are not allowed in
the material definition.
Temperature and field variable dependencies are allowed to define the elasticity of
the matrix material only, and only one inclusion is allowed in this case.
Because the multiscale material is replaced by a composite with homogenized viscoelastic
properties, the microlevel output is not available.
Composites with Damage
When damage and failure are defined at the constituent level, the damage in
each constituent contributes to the overall damage in an indirect way through
stress averaging and strain partitioning. As the stiffness of the constituent
decreases, the strain increment in this constituent is likely to increase based
on the formulation of the homogenization. In reality, the damage behavior of
each constituent is likely to interact with each other and affect the overall
damage behavior of the composite; therefore, you might need to specify an
additional damage variable, ,
for the composite. At any time during the analysis, the stress tensor of the
composite is given by
where , and is the effective stress of the constituent.
You can choose to combine the damage variables, ,
in all constituents to get the overall damage behavior of the composite. The
overall damage variable, ,
can be computed as the maximum of :
or you can choose to combine the damage variables in a multiplicative sense:
You can also specify
using user subroutine
UDAMAGEMF. In addition, you can specify a separate damage variable,
,
for each constituent. This variable is used to account for the additional
contribution to the overall damage behavior due to interaction (for example, a
fiber material might buckle more easily if cracks start to develop in the
matrix material) as follows:
Example
This simple example illustrates how to define a multiscale material.
The name of the multiscale material is
COMPOSITE. The material consists of one inclusion
material
MAT2 embedded in the matrix material
MAT1. The shape of the inclusion is prolate. The
Mori-Tanaka homogenization method is used. The inclusion has a volume fraction
of
and an aspect ratio of .
The direction of the inclusion is fixed and defined with a vector
.
The name of the matrix is
MATRIX_MAT1, and the name of the inclusion is
FIBER_MAT2.
In Abaqus/Standard the multiscale material model can be used with
three-dimensional solid elements, including stress/displacement elements, diffusive
heat transfer elements, and coupled temperature-displacement elements;
two-dimensional solid elements (stress/displacement only);
membrane elements;
three-dimensional conventional shell elements (stress/displacement only); and
continuum shell elements (stress/displacement only).
In Abaqus/Standard additional plane stress iterations are included for two-dimensional plane stress elements
and shell elements.
In Abaqus/Explicit the multiscale material model can be used only with three-dimensional solid
stress/displacement elements.
Hybrid elements and incompatible mode elements are not supported.
Output
By default, the output variables are macrolevel results. In addition, you can also request
microlevel output. The name of the constituent is appended to the end of the output
variable.
For composites reinforced with dispersed short fibers, the fiber output variables are
computed in the local system with the local fiber direction as the maximum principal
direction of the orientation tensor. The matrix output is available only if the direct
Mori-Tanaka homogenization approach is used. Otherwise, only averaged stress and strain are
available. If plasticity is specified in the matrix material, averaged plastic strains are
also available. The same matrix output limitation also applies to composites reinforced with
multiple fiber families.
Microlevel strain output is not available in Abaqus/Explicit.
References
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