Solution Manual
Solution to Suggested Problems
Problem 10.1 Using a three-dimensional Finite element model and the Modified Virtual Crack
Closure Technique (MVCCT, [79]) compute the ERR distribution along the straight delamination
front for the DCB problem in Figures 10.14 and 10.15. Compare these results with those from the
interface model (details can be found in [60]). Use the same mesh adopted for results of Figure
10.15 in the x-y plane and one row of solid finite element in the thickness direction.
The results from the interface model must show, away from the edges, a satisfactory accuracy in
comparison with those obtained by a three-dimensional formulation, although each sublaminate is
assumed as composed by simply one shear deformable plates.
This confirms the results obtained in the 2D delamination problem, where it was proved that for
special loading and geometrical schemes a double plate model suffices to determine accurately
the mode mix. It is expected that for more general loading and geometry conditions (mixed-mode
loadings, offset delaminations, for instance) more than one plate is needed to model each sublaminate to capture the actual 3D character of the problem near the delamination front.
Solution to Problem 10.1
The suggested problem 10.1 involves a mode II loading condition for a three-dimensional DCB
geometry. The properties of the plates are as follows
a = 256 mm, h=16 mm, B = 400 mm, L = 512 mm, E = 80.00 GPa, = 0.3, N =6.25 N/mm
The size of the elements at the delamination front is 1 mm in the x-direction and 8 mm in the ydirection. The maximum ratio between the x and y dimensions is 8 for all the elements. The
interface stiffness parameters for the normal and tangential directions is chosen equal to 108
N/mm3 according to the numerical analyses carried out to investigate the sensitivity of the results
with respect to the adopted value of the penalty parameter.
The commented ANSYS command file containing the command sequence for this example is
available at the web resource: http://www.mae.wvu.edu/barbero/feacm/ (Problem_10_1.lgw). To
run the example, go to the File menu and select "Read input from". The SOLID5 element type with
eight node is used to mesh the solid layers of the delaminate plate model, whereas COMBIN14
element type are used for the interface model. At the end of the analysis the log file produces four
text files,
namely G_I_prob_10_1.txt, G_II_prob_10_1.txt, G_III_prob_10_1.txt and
Y_I_prob_10_1.txt, containing respectively the numerical values of normalized ERRs distributions
and the corresponding y-coordinate along the delamination front.
Interface elements, necessary to connect the two sub-laminates in the region -D, are
implemented by using three spring elements (COMBIN14), each one acting in a different
translational direction, for each pair of nodes of the upper and lower layers of solid finite elements
placed in correspondence of the undelaminated plane. For simplicity the linear elastic interface has
been modelled only along the delamination front, whereas the undelaminated portion of the
interface has been modelled by means of three constraint equations between each pair of nodes
relative solid finite elements of the upper and lower layers contained in the undelaminated plane.
Nodes and elements of the mesh are generated by means of the APDL *DO command.
It is worth noting that for the prevalent mode II loading condition it suffices to model only x and ytranslational springs and constraint equations. However, in order to provide a general code to be
used for general mixed-mode loading conditions, a complete modelling of interface elements has
been implemented. The deformed mesh of the solid FE model is shown in figure 10.1.
1
Z
Y
X
Problem 10.1
Fig. 10.1. Deformed mesh of the solid FE model for the problem 10.1
The ERR distributions along the straight delamination front are evaluated by using the MVVCT,
which leads to a procedure similar to that expressed in equations (10.34), and are illustrated in
Figures 10.14 and 10.15. Comparison with results using (1-1) and (2-2) plate assemblies in
conjunction with interface elements are also shown in Figure 10.14 and 10.15.
The mode II and III ERR distributions along the delamination front have been normalized with
respect to the approximate values GII2D (see expression 10.35 introduced in the solution to
example 10.15) equal to 6.943 10-6N/mm.
Problem 10.2 Use ANSYS to analyze a double plate assembly comprising two orthotropic layers
with different thicknesses, and subjected to a pair of opposed opening forces as shown in Figure
10.18.
The upper sublaminate has the following properties:
E1x = 35,000 MPa, E1y = E1z =10,500, G1yz = 1,050 MPa, G1xy = G1xz = 1,167, 1xy =1xz
=1yz = 0.3.
The lower sublaminate has the following properties:
E2x = 70,000 MPa, E2y = E2z =21,000, G2yz = 2,099.7 MPa, G2xy = G2xz = 2,333, 2xy = 2xz =
2yz = 0.3.
The opening force T is equal to 1 N/mm and the dimensions of the laminate are: a=10 mm, B=20
mm, L=20 mm, h1 =0.5 mm, h2=1 mm.
Use mesh refinement at the delamination front similar to that used in Example 10.5, in which the
maximum ratio between the x and y dimensions is 8. Use interface stiffness k z and kxy equal to
108 N/mm3 and the (2-4) model, which is sufficiently accurate since this subdivision reflects plate
geometry by using more plate elements for the thicker sublaminate. Calculate the mode I, II and III
components of ERR along the delamination front. Results may be found in [61].
Solution to Problem 10.2
The commented ANSYS command file containing the command sequence for this example is
available at the web resource: http://www.mae.wvu.edu/barbero/feacm/ (Problem_10_2.lgw). To
run the example, go to the File menu and select "Read input from".
The SHELL181 element type is used to mesh the plate models, whereas COMBIN14 element type
are used for the interface model. The same methodology of Example 10.3 is used to connect
COMBIN14 elements to the mid-plane plate nodes. Displacement continuity at the interfaces
between sublaminates is imposed by means of constraint equations.
Interface elements, necessary to connect the two sub-laminates in the region -D, are
implemented by using a combination of rigid links (ANSYS MPC184, defined by two nodes and
three degrees of freedom at each node) and spring elements (ANSYS COMBIN14). Spring
elements are placed among the offset nodes. The offset nodes are generated by CEs. For
simplicity the linear elastic interface has been modelled only along the delamination front, whereas
the undelaminated portion of the interface has been modelled only by means of constraint
equations.
For each mid-plane node of the upper plate model, three coincident nodes located on the lower
surface of the plate are created, and constrained to deform as embedded in a plate segment
normal to the mid-plane by means of constraint equations. Similarly, for each mid-plane node of
the lower plate model, three coincident nodes located on the upper surface are created. Three
COMBIN14 elements connected to the three pairs of coincident nodes placed at the delamination
plane are then introduced, each one acting in a different translational direction.
At the end of the analysis the log file produces five text files, namely G_I_prob_10_2.txt,
G_II_prob_10_2.txt, G_III_prob_10_2.txt, G_tot_prob_10_2.txt and Y_I_prob_10_1.txt, containing
respectively the numerical values of normalized ERRs distributions and the corresponding ycoordinate along the delamination front.
Nodes and elements of the mesh are generated by means of the APDL *DO command.
The deformed mesh of the solid FE model is shown in figure 10.2.1.
1
Z
Y
X
10.2.1. Deformed mesh of the (2-4) multi-layer plate FE model for the problem 10.2
Results shown in figure 10.2.2 taken from [61] show that mode I and II components of energy
release rate are predominant, although near the free edges a small amount of mode III appears.
Fig. 10.2.2. Normalized distributions of total energy release rate and its
individual components as a function of the position along the delamination
front taken from [61].
Problem 10.3 Solve the example of Problem 10.2 by using solid finite elements and perform mesh
refinement analysis to evaluate the behavior of the total ERR and its mode components when the
size of the delamination front element in the x-direction decreases.
Consistently with the (2-4) multi-layer modeling, two and four layers of solid finite elements can be
placed in the thickness direction for the upper and lower sublaminates, respectively. The laminate
geometry and the 3D FE model are illustrated in Figure 10.19b. The ERR mode components for
the 3D model must be calculated by using the MVCCT [79] which leads to an expression similar to
(10.34), by evaluating the node forces as Lagrange multipliers related to the adhesion constraints
along the un-delaminated region of the delamination plane.
The analysis should show that while both the individual components and the total ERR converge
as the delamination front elements are smaller for the multi-layer plate model, on the contrary,
results obtained by using solid finite elements should show a non-convergence behavior for the
individual ERR. Results may be found in [61]. This is a consequence of the mismatch of material
properties across the interface which leads to an oscillatory singularity behavior of stresses and
displacements near the delamination front [41], in place of the inverse square-root singularity which
occurs when delamination is placed between two equal orthotropic or isotropic layers, as is the
case of the examples analyzed in the previous sections. Therefore, the proposed method can be
used as a computationally efficient method to eliminate the oscillatory singularity that causes nonconvergent behavior when solid finite elements are used.
Solution to Problem 10.3
The commented ANSYS command file containing the command sequence for this example is
available at the web resource: http://www.mae.wvu.edu/barbero/feacm/ (Problem_10_3.lgw). To
run the example, go to the File menu and select "Read input from".
The SOLID45 element type with eight node is used to mesh the solid layers of the delaminate plate
model, whereas COMBIN14 element type are used for the interface model. At the end of the
analysis the log file produces four text files, namely G_I_prob_10_1.txt, G_II_prob_10_1.txt,
G_III_prob_10_1.txt and Y_I_prob_10_1.txt, containing respectively the numerical values of
normalized ERRs distributions and the corresponding y-coordinate along the delamination front.
Interface elements, necessary to connect the two sub-laminates in the region -D, are
implemented by using three spring elements (COMBIN14), each one acting in a different
translational direction, for each pair of nodes of the upper and lower layers of solid finite elements
placed in correspondence of the undelaminated plane. For simplicity the linear elastic interface has
been modelled only along the delamination front, whereas along the undelaminated portion of the
interface displacement continuity is ensured by using coincident nodes for those solid elements
adjacent to the undelaminated plane. Nodes and elements of the mesh are generated by means of
the APDL *DO command.
In order to perform a convergence analysis the ANSYS command file by means of the *ASK
command prompts the user to input the value of the delamination front element size (x) which
must assume one of the following values: 0.1mm, 0.04mm, 0.025mm, 0.016mm, 0.0125mm. As a
matter of fact, in a zone near the delamination front the mesh is organized in such a way to change
the size of finite elements in the y-direction according to the size in the x-direction, in order to
ensure that the maximum ratio between the x and y dimensions is 8.
At the end of the analysis the log file produces five text files, namely G_I_prob_10_2.txt,
G_II_prob_10_2.txt, G_III_prob_10_2.txt, G_tot_prob_10_2.txt and Y_I_prob_10_1.txt, containing
respectively the numerical values of normalized ERRs distributions and the corresponding ycoordinate along the delamination front, for the chosen value of x. Therefore the convergence
analysis may be performed by running the ANSYS command file for decreasing values of x.
The deformed mesh of the solid FE model is shown in figure 10.3.1.
1
Z
Y
X
Fig. 10.3.1. Deformed mesh of the solid plate FE model for the problem 10.3.
Using delamination front element sizes equal to: x/h1 = 0.2, 0.08, 0.05, 0.032, 0.025 and
representing the energy release rates behaviour for different points at the delamination front (y/B =
0.02, 0.06, 0.1, 0.5) leads to figures 10.3.2, 10.3.3 and 10.3.4. The individual energy release rate
components show a non convergent behaviour while the total energy release rate achieves rapidly
an asymptotic value and agrees well with the multi-layer plate results. A more detailed
convergence analysis can be found in Figs 16, 17, and 18 of reference [61].
14000
y/B=0.5
12000
GE2h2/T2
y/B=0.1
y/B=0.06
10000
8000
y/B=0.02
6000
1.00
0.10
0.01
x/h1
Fig. 10.3.2. Total energy release rate at different points along the
delamination front as the delamination front element size decreases. A
logarithmic scale is used for the x axis.
10000
y/B=0.5
9000
y/B=0.1
y/B=0.06
GIE2h2/T
2
8000
7000
6000
5000
y/B=0.02
4000
1
0.1
x/h1
Fig.10.3.3. Mode I energy release rate at different point along the
delamination front as the delamination front element size decreases. A
logarithmic scale is used for the x axis.
0.01
3300
y/B=0.1
3100
y/B=0.5
GIIE2h2/T2
2900
y/B=0.06
2700
2500
2300
2100
y/B=0.02
1900
1700
1
0.1
x/h1
Fig. 10.3.4. Mode II energy release rate at different point along the
delamination front as the delamination front element size decreases. A
logarithmic scale is used for the x axis.
0.01