3.1.6.15. Series3D Material Wrapper¶

This command is used to construct a Series3D material object. It is a wrapper that imposes an iso-stress condition to an arbitrary number of previously-defined 3D nDMaterial objects

3.1.6.15.1. Theory¶

This model imposes a minimal kinematic constraint on the sub-materials such that the macro-scopic strain tensor $\varepsilon_{m}$ (i.e. the strain tensor of the wrapper Series3D material) is equal to the volumetric average of the micro-scopic strain tensors $\varepsilon_{i}$ (i.e. the strain tensors of each sub-material)

(3.1.6.1)¶\[\varepsilon_{m} = \frac{1}{V} \int_{V} \varepsilon_{i} \,dV\]

The above equation can be rewritten as a weighted sum

(3.1.6.2)¶\[\varepsilon_{m} = \sum_{i=1}^{n} \varepsilon_{i}w_{i}\]

where the weight $w_{i}$ is the volume fraction of the i^th sub-material. Imposing the constraint in (3.1.6.2) is equivalent to impose forces on each sub-material proportional to each material’s volume fraction:

(3.1.6.3)¶\[F = C\begin{bmatrix} w_1 & w_2 & ... & w_n \end{bmatrix}\]

But the force acting on each volume fraction (with its own sub-material) is

(3.1.6.4)¶\[F_i = V\sigma_iw_i\]

therefore, the following equality can be written

(3.1.6.5)¶\[F = C\begin{bmatrix} w_1 & w_2 & ... & w_n \end{bmatrix} = V\begin{bmatrix} \sigma_1w_1 & \sigma_2w_2 & ... & \sigma_nw_n \end{bmatrix}\]

which shows that (3.1.6.2) actually imposes an iso-stress condition on each sum-material

(3.1.6.6)¶\[\frac{C}{V} = \sigma_m = \sigma_1 = \sigma_2 = ... = \sigma_n\]

nDMaterial Series3D $matTag $tag1 $tag2 ... $tagN <-weights $w1 $w2 ... $wN> <-maxIter $maxIter> <-relTol $relTol> <-absTol $absTol> <-verbose>

Argument	Type	Description
$matTag	integer	unique tag identifying this series material wrapper
$tag1 $tag2 … $tagN	N integer	unique tags identifying previously defined nD materials
$w1 $w2 … $wN	N float	weight factors, optional. If not defined, they will be assumed all equal to 1
-maxIter	string	string keyword to specify a user-defined maximum number of iterations
$maxIter	integer	maximum number of iterations to impose the iso-stress condition, optional, default = 10
-relTol	string	string keyword to specify a user-defined relative stress tolerance for the iso-stress condition
$relTol	float	relative stress tolerance for the iso-stress condition, optional, default = 1.0e-4
-absTol	string	string keyword to specify a user-defined absolute stress tolerance for the iso-stress condition
$absTol	float	absolute stress tolerance for the iso-stress condition, optional, default = 1.0e-8
-verbose	string	string keyword to activate print of debug information

3.1.6.15.2. Usage Notes¶

Limitations

The only material formulation for the Series3D material object is “ThreeDimensional”.
The only material formulation allowed for the sub-material objects is “ThreeDimensional”.

Responses

All responses available for the nDMaterial object: stress (or stresses), strain (or strains), tangent (or Tangent), TempAndElong.
material $matId … : use the material keyword followed by the 1-based index of the sub-material (and followed by the desired response) to forward the request to the matId sub-material.
homogenized … : use the homogenized keyword followed by the desired response to forward the request to all sub-materials, and to compute its weighted average.

Example 1 - Simple Linear Validation

A simple example to validate the Series3D material. First material is twice as stiff as the second one. All weights are assumed equal to 1.

The expected results are:

equal stress
additive strain
strain in the soft material twice as large as the strain in the stiff material

Tcl Code

# the 2D model
wipe
model basic -ndm 2 -ndf 2

# 2 young's moduli
set E1 30000.0
set E2 [expr $E1*0.5]

# 2 elastic materials
nDMaterial ElasticIsotropic 1 $E1 0.2
nDMaterial ElasticIsotropic 2 $E2 0.2

# the Series3D wrapper using all weights = 1
nDMaterial Series3D 3   1 2

# a triangle
node 1 0 0
node 2 1 0
node 3 0 1
nDMaterial PlaneStress 100 3
element tri31 1   1 2 3   1.0 "PlaneStress" 100

# fixity
fix 1   1 1
fix 2   0 1
fix 3   1 0

# a simple ramp
timeSeries Linear 1

# imposed macroscopic strain in XX component
set em 0.01
pattern Plain 1 1 {
  sp 2 1   $em
}

# solve
constraints Transformation
numberer Plain
system FullGeneral
test NormDispIncr 1.0e-6 10 0
algorithm Newton
integrator LoadControl 1.0
analysis Static
analyze 1

# check responses
puts "Checking responses"
set Sm [expr [lindex [eleResponse 1 material 1 stress] 0]]
set S1 [expr [lindex [eleResponse 1 material 1 material 1 stress] 0]]
set S2 [expr [lindex [eleResponse 1 material 1 material 2 stress] 0]]
set SmHom [expr [lindex [eleResponse 1 material 1 homogenized stress] 0]]
puts "Sm = S1 = S2 = SmHom -> [format {%6.3f = %6.3f = %6.3f = %6.3f} $Sm $S1 $S2 $SmHom] (stresses are equal)"
set Em [expr [lindex [eleResponse 1 material 1 strain] 0]]
set E1 [expr [lindex [eleResponse 1 material 1 material 1 strain] 0]]
set E2 [expr [lindex [eleResponse 1 material 1 material 2 strain] 0]]
puts "Em = E1 + E2 -> [format {%6.5f = %6.5f + %6.5f} $Em $E1 $E2] (strains are additive since w1=w2=1)"

Code Developed by: Massimo Petracca at ASDEA Software, Italy.