'''Viscoelastic Homogenization'''

In this example, we want to compute the effective properties of a composite material made of isotropic viscoelastic matrix and transversely isotropic elastic fiber. The fiber properties are defined by means of engineering constants as specified in the table below.

[[Image(Fiber_properties.png, desc="Fiber properties defined as transversely isotropic elastic")]]

The resin properties are given by means of the Prony coefficients presented in the table below and following the formulation of Figure 1. In addition, we will consider that the resin has a constant Poisson's ratio equal to 0.33.

[[Image(Formula_Young.png, desc="Definition of the Young modulus of the resin")]] 

[[Image(Resin_properties.png, desc="Prony coefficients and relaxation times for the resin")]]

We will use a square pack 2D SG with fiber volume fraction equal to v,,f,, = 0.64.


====Software Used==== 
In his tutorial we will use Abaqus CAE with the Abaqus SwiftComp GUI  plug-in. Abaqus CAE will be used to GUi to define the time-dependent material properties and to run the viscoelastic homogenization. SwiftComp will run in the background. 

====Solution Procedure==== 
The steps required to compute the effective viscoelastic properties using Abaqus SwiftComp GUI are as follows. 

''' # Step 1.''' We define the material properties in global coordinate system. We click on ''Materials'' in Abaqus CAE and we define the  ''Fiber'' properties by means of the engineering constants.
[[Image(Step_1.png, Step_2_A1.png desc="Definition of the fiber properties")]] 

''' # Step 2.''' From the default the Abaqus SwiftComp GUI SGs, we pick the 2D Structure Genome with Hexagonal pack. 

[[Image(Step_2.png, desc="Definition of the 2D SG hexagonal pack microstructure")]] 

''' # Step 3.''' Now, in order to compute the homogenized thermal conductivity properties, we click on ''Homogenization'' and we select ''Conduction'' in Analysis Type. 

[[Image(Step_3.png, desc="Definition of the homogenization step")]] 

''' # Step 4.''' We click on ''Ok'' to run the homogenization step. SwiftComp on the background will run the homogenization. 

[[Image(Step_4.png, desc="SwiftComp running on the background")]] 

''' # Step 5.'''The results can be found in the ''.sc.k'' file as shown next. Note that the first matrix corresponds to the effective thermal conductivity matrix in the form of K,,ij,,^*^. The second matrix corresponds to the compliance matrix in the form of (K,,ij,,^*^ )^-1^.

[[Image(Step_5.png, desc="Results corresponding to the effective thermal conductivities")]] 

====References====
 # Rique, O.; Barocio, E.; Yu, W.: “Experimental and Numerical Determination of the Thermal Conductivity Tensor for Composites Manufacturing Simulation,” ASC 32nd Technical Conference, October 2017, Purdue University.