HeliusMCT v2 Example Problem 1 Abaqus

Example Problem 1 Helius:MCT™ Version 2.0 for Abaqus July, 2009 Abstract The objective of this example problem is to demonstrate the modeling process for conducting a failure analysis of a large composite structure using Helius:MCT. This example will demonstrate the effects of through-the-thickness mesh density, finite element type, and failure criteria type on the progressive failure response of the composite structure. For questions, comments or further information, contact Firehole Technologies at [email protected] Legal Notices Copyright 2009, Firehole Technologies, Inc. Helius:MCT is a trademark of Firehole Technologies, Inc. Any use of the Helius:MCT trademark requires the prior written consent of Firehole Technologies, Inc. Abaqus/Standard is a trademark of Dassault Systemes S.A. and Dassault Systemes SIMULIA Corp.

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Table of Contents E1.1 Introduction ......................................................................................................................................... 3 E1.2 Problem Description ............................................................................................................................ 4 E1.3 Modeling Comparison ......................................................................................................................... 7

E1.3.1 Linear Elastic Analysis ................................................................................................................. 7 E1.3.2 First Failure Analysis ................................................................................................................... 9 E1.3.3 Effect of Through-Thickness Mesh Density on Progressive Failure Response ......................... 11 E1.3.4 Effect of Element Type on Progressive Failure Response ......................................................... 14

E1.4 Comparison of Helius:MCT with currently available Abaqus failure criteria .................................. 17 E1.4.1 Comparison of Helius:MCT with linear elastic Abaqus failure criteria. .................................... 17 E1.4.2 Comparison of Helius:MCT with Abaqus’ progressive damage model ..................................... 20

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Helius:MCT™ Example Problem 1

Composite Conic

E1.1 Introduction The objective of this example problem is to demonstrate the modeling process for conducting a failure analysis of a large composite structure using Helius:MCT (herein referred to as Helius). This example will demonstrate the effects of through-the-thickness mesh density, finite element type, and failure criteria type on the progressive failure response of the composite structure. For detailed instructions on how to use Helius, please refer to the Helius:MCT User’s Guide and Tutorials 1 and 2. In a typical composite structure, catastrophic global structural failure is precipitated by the onset and subsequent growth (or spreading) of localized matrix and fiber constituent failures. Consequently, three different types of failure are discussed in this example problem.

1. Local Failure (failure at a Gaussian integration point):

A. Matrix Constituent Failure B. Fiber Constituent Failure

2. Global Failure (significant, discrete reduction in global stiffness of the structure): This example problem involves imposing a set of quasi-static monotonically increasing loads on a large composite structure. The overall response of the composite structure to these applied loads is characterized by the displacement at certain points on the structure. Global structure failure is identified by the appearance of a significant discontinuity in the structure's load/displacement relationship indicating a large decrease in the overall stiffness of the structure.

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E1.2 Problem Description The example problem consists of a conical, composite sandwich structure that is subjected to a quasi-static axial compressive load that is monotonically increased until global structure failure occurs. The example problem is similar to the load-controlled tests that are part of the composite structure’s flight qualification testing. Figure 1 shows the finite element mesh used to represent the conical composite structure, the load head and the load head adapter.

Figure 1: Test Assembly

Four actuators are used to apply concentrated vertical compressive forces to a load head that sits on top of the conic structure (Figure 1). Table 1 lists the compressive forces applied to each of the four positions on the load head. As seen in Table 1, these forces result in a uniform vertical compressive load where 100% loading corresponds to a total vertical compressive force of 200,000 lbs.

270°

180°

90°

0°

Conical Sandwich Composite Structure

Load Head Adapter

Load Head

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Table 1: Load scheme (uniform compression)

000 090 180 2700% 0 0 0 0 0

50% -250 -250 -250 -250 -1000100% -500 -500 -500 -500 -2000

- Loads are in kips- Loading is quasi-static and is linearly ramped up- Negative sign indicates a compressive force

Actuator AzimuthLoad Level Total Load

An adapter is used to connect the load head to the conic structure (Figure 1). For simplicity, the adapter and load head are considered rigid when compared to the composite conic, therefore, an arbitrarily large elastic modulus was assigned to the isotropic material used for both the load head and the adapter. As seen in Figure 1, an access door is cut through the side of the conic structure. The boundary conditions for the conic structure are provided by constraining the entire bottom surface of the conic structure using fixed boundary conditions (displacements in the 1, 2 and 3 directions are constrained to have zero displacement). The composite conic structure features a sandwich panel construction. The through-the-thickness profile of the sandwich panel (Figure 2) is uniform in both the hoop and axial directions on the conic. The inner and outer composite facesheets of the sandwich construction have a [(90/0)4] layup as measured from the inside surface (0° is in the axial direction, 90° in the hoop direction). Each of the composite plies is 0.0075” thick and composed of carbon/epoxy AS4-3501-6. The 1” thick core is an isotropic foam material (Rohacell 110 WF).

8 plies 0.06"

8 plies 0.06"

1.0" 1.12"Core - Rohacell 110 WF

Inner Facesheet [(90/0)4] - AS4-3501-6

Outer Facesheet [(90/0)4] - AS4-3501-6Outer (Bag) Surface

Inner (Tooling) Surface

Figure 2: Sandwich panel profile

Material properties for AS4-3501-6 and Rohacell 110 WF are provided in Figure 3. Helius:MCT assumes all composite lamina are transversely isotropic, so a transversely isotropic material model will be used for AS4-3501-6.

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E11 1.83E+07 E 2.61E+04

E22 = E33 1.62E+06 ν 0.286

ν12 = ν13 0.282

ν23 0.545

G12 = G13 9.51E+05

G23 5.25E+05

S11+ 2.83E+05

S11- 2.15E+05

S22+ = S33

+ 6.96E+03

S22- = S33

- 2.90E+04

S12 = S13 1.15E+04S23 7.25E+03

Property TableAS4-3501-6

[psi ]Rohacell_110WF

[psi ]

Figure 3: Material properties for AS4-3501-6 and Rohacell_110WF

Figure 4 provides the dimensions for the composite conic structure.

Figure 4: Schematic for cross-section of composite conic (dimensions are in inches)

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E1.3 Modeling Comparison A series of different finite element models was created to demonstrate the effect of three key modeling issues on the predicted failure response of the composite structure:

• Number of elements distributed through the thickness of the sandwich laminate • Type of finite element • Type of failure criteria

Note: In this study, the surface mesh density of all finite element models remained constant.

E1.3.1 Linear Elastic Analysis Abaqus version 6.8 provides a three-dimensional continuum element (e.g., C3D8R) that can utilize a multilayer composite lay-up. Previous versions of Abaqus did not have this functionality. The use of continuum elements that explicitly account for all six stress components (σ11, σ22, σ33, σ12, σ13, σ23) ensures that the material failure criteria can utilize the transverse stress components (σ33, σ13, σ23) that are neglected (or only approximated) in conventional or continuum shell elements. To demonstrate the importance of explicitly accounting for the transverse stress components, a simple linear elastic analysis is performed to allow comparison of the magnitudes of the in-plane and transverse stress components in the most highly stressed region of the model. Model(s) linear_elastic_solid.inp

• Element type: C3D8R • Through-thickness mesh density:

o Composite Facesheets: 1 element o Foam Core: 1 element

Results Figure 5 displays the stress state for a point in the outside surface ply near the upper left-hand corner of the access door of the composite conic. The fiber constituent failure criterion is dependent on three stress components (σ11, σ12, σ13). Examination of Figure 5 reveals that the transverse shear stress (σ13) is negligible in comparison to the in-plane stresses (e.g., σ13 is only 0.5% of σ11). This indicates that transverse stresses will not provide a significant contribution to fiber failure. On the other hand, matrix constituent failure is driven by all six stress components. By comparing the magnitude of the out-of-plane stresses (σ33, σ13, σ23) with the magnitude of the in-plane stresses (σ22, σ12), we can see that the out-of-plane stresses will make a significant contribution to the matrix failure criterion. In order to accurately predict failure in the structure, matrix failure must be captured properly, because after a matrix failure event, its stiffness is reduced and stress redistribution causes an increased stress state in the fibers. Because the transverse stresses are not negligible in certain regions of large composite structures, Firehole Technologies recommends that solid elements should be used wherever possible. This will be demonstrated when solid elements are compared with continuum shell elements later in this documentation (Section E2.2.4).

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Value Normalized Normalized Normalized[psi] (σij / σ11) (σij / σ22) (σij / σ12)

σ11 267,057 100.0% 1045.6% 1575.6%

σ22 25,540 9.6% 100.0% 150.7%

σ33 3,623 1.4% 14.2% 21.4%

σ12 16,949 6.3% 66.4% 100.0%

σ13 1,274 0.5% 5.0% 7.5%σ23 932 0.3% 3.6% 5.5%

Stress Component

Figure 5: Stress state of selected element

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E1.3.2 First Failure Analysis As a first attempt at a progressive failure analysis, a through-thickness mesh density of 1 element per facesheet and 1 element for the core, for a total of 3 elements through the thickness of the sandwich construction is used. In this case, the C3D8R (reduced integration element) is used. Note that it is possible to use a single element through the thickness of the entire laminated sandwich structure; however, due to the relatively high compliance of the thick foam core, the use of a single element through the laminate thickness yields very inaccurate results. Model(s) 1elemFace_1elemCore.inp


o Composite Facesheets: 1 element o Foam Core: 1 element

Results Table 2 displays the load level at which each type of failure event is predicted. The criteria used to predict localized matrix failure and localized fiber failure are described in detail in the Helius:MCT Theory Manual. In Table 2, it should be emphasized that the matrix failure and fiber failure load levels indicate the load at which a localized matrix or fiber failure is first detected. Note that this first instance of matrix failure or fiber failure occurs at a single Gaussian integration point within one of the material plies of one of the elements of the model. In a large composite structure containing thousands (or millions) of Gaussian integration points, a very large number of localized constituent failures are necessary in order to detect any appreciable change in the overall stiffness of the composite structure. Global structural failure can be defined in many different ways, but for the purpose of this example problem, global failure is defined as a large discontinuity in the composite structure’s overall vertical load-displacement curve. The overall vertical deformation of the composite structure is quantified by using the vertical displacement at the load application point labeled 0° in Figure 1. Because the load head and adapter are considered rigid, a large discontinuity in the load-displacement curve is indicative of very rapid growth (spreading) of localized material failures that occur during a particular load increment, resulting in a large degradation of the overall stiffness of the composite structure. This definition of global structural failure is chosen since most experimental tests are stopped at this point to prevent damage to expensive test equipment. Figure 6 shows the overall vertical load-deflection curve for the composite structure. Note that the overall response of the structure appears to be linear up to a load level of 56%, at which time a global structural failure occurs. As seen in Table 2, the first localized matrix failure occurred at a load of 49% and the first localized fiber failure occurred at a load of 49%. However, all of the localized failures that occurred between the load range of 49% and 56% were not sufficient to produce a visually detectable change in the composite structure’s overall load-deflection response. When the load is increased from 56% to 57%, a very large cascade of localized failures occurs. This failure cascade is significant enough to destroy approximately 92% of the vertical stiffness of the composite structure (the vertical displacement increases by a factor of 12 during the load increment). This type of behavior, where the overall response of the composite structure remains approximately linear up to global structural failure is fairly common among structures composed of brittle composite materials.

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Table 2: Failure events for composite conic

# Elements # Elements(facesheet) (core)

1elemFace_1elemCore C3D8R 1 1 49% 49% 57%

Notes: 1: Load percentage is on a 0-100% scale

2: Global failure is defined as a large discontinuity in the load-vertical displacement curve for the 0° application point of the load head (Figure 1)

Model Element Type Matrix Failure1 Fiber Failure1 Global Failure1,2

Figure 6: Vertical load-displacement curve for the 0° actuator point This first failure analysis is simply intended to provide a base-level prediction that can be used for comparing the results from subsequent modeling efforts.

Global Failure

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E1.3.3 Effect of Through-Thickness Mesh Density on Progressive Failure Response The progressive failure response of the composite structure was predicted using four finite element models that differed only in the number of elements distributed through the thickness of the sandwich panel, i.e., the materials, element type, in-plane mesh density and boundary conditions were exactly the same in each model. The four different levels of through-the-thickness mesh density are listed below. Each of the four models uses C3D8R elements.

• Original model: 1 element to capture core and 1 element per facesheet that captures the entire layup (3 elements through-thickness).

• 1st modification: Increase the core mesh density to allow 4 elements to capture the core profile (6 elements through-thickness).

• 2nd modification: Increase the facesheet density to allow 2 elements to capture the layup of each facesheet (8 elements through-thickness).

• 3rd modification: Increase the facesheet density to allow 4 elements to capture the layup of each facesheet (12 elements through-thickness).

Ideally, a very fine through-thickness mesh would be utilized to provide the most accurate loading distribution and deformation throughout the model, but run time costs often limit mesh densities. Model(s) 1elemFace_4elemCore.inp


o facesheets: 1 element o core: 4 elements

2elemFace_4elemCore.inp


o facesheets: 2 elements o core: 4 elements

4elemFace_4elemCore.inp



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Results Table 3 displays the load level at which each type of failure event is predicted by the four models. Figure 7 displays the overall vertical load-displacement curves for the composite structure.



1elemFace_1elemCore C3D8R 1 1 49% 49% 57%1elemFace_4elemCore C3D8R 1 4 48% 55% 64%2elemFace_4elemCore C3D8R 2 4 52% 52% 60%4elemFace_4elemCore C3D8R 4 4 59% 60% 60%




Figure 7: Vertical load-displacement curve for the 0° actuator point

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Comparing mesh density results, the following observations result:

1. As the through-the-thickness mesh density is increased, the initiation of localized matrix and fiber failure is predicted to occur at higher load levels. - As we progressively increase the number of elements through the thickness of the laminate,

the transverse shear stiffness of the model decreases faster than the in-plane stiffness of the model. Consequently, as the through-the-thickness mesh density is increased, the model tends to exhibit an increase in transverse shear deformation at the expense of in-plane deformation. The net result of this trend is that the magnitude of the peak in-plane stresses tends to decrease as the through-the-thickness mesh density is increased, thus localized failure is predicted to occur at higher load levels.

2. Global failure is only slightly affected by mesh density.

- Despite the fact that increasing the through-the-thickness mesh density results in delayed

local failure initiation for both the matrix and the fiber, the predicted global structural failure load is not significantly affected by the through-the-thickness mesh density. This observation suggests that the escalation from local failure initiation to global structural failure occurs faster as the through-the-thickness mesh density is increased. It is reasonable to expect local failures to progress into global failures more rapidly for a higher density mesh (as seen in Table 3). A denser mesh results in a more refined failure path for a structure due to the increase in number of Gaussian integration points for failure criterion to be evaluated at. A more refined failure path facilitates a more rapid progression of failure, hence the rapid progression of local failures into global failures.

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E1.3.4 Effect of Element Type on Progressive Failure Response The progressive failure response of the composite structure was predicted using three finite element models that differed only in the type of element; i.e., the materials, mesh density and boundary conditions were exactly the same in each model. Three Abaqus element types were tested: C3D8R, C3D8 and SC8R. Each of the three models used the same through-the-thickness mesh density, namely, four elements were used for the foam core and each composite facesheet was divided into two elements for a total of eight elements through the thickness of the sandwich laminate. Model(s) C3D8.inp

• Element type: C3D8 • Through-thickness mesh density:

o Each composite facesheet: 2 elements o Foam Core: 4 elements

SC8R.inp

• Element type: SC8R • Through-thickness mesh density:

o Each composite facesheet: 2 elements o Foam Core: 4 elements

Results Table 4 displays the load level at which each type of failure event is predicted by the three models. Figure 8 displays the overall vertical load-displacement curves for the composite structure predicted by the three models.



2elemFace_4elemCore C3D8R 2 4 52% 52% 60%C3D8 C3D8 2 4 41% 47% 55%SC8R SC8R 2 4 42% 43% 68%




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Comparing reduced integration continuum elements with fully integrated continuum elements (C3D8R vs. C3D8), the following observations and explanations are given:

1. A fully integrated element type results in local failure events (matrix and fiber failure) occurring at lower load levels.

- The fully integrated element utilizes more Gaussian integration points than the reduced integration element. Even if both elements predict the same element average stresses, the fully integrated element will predict a higher local peak stress than the reduced integration element simply because it has more Gaussian integration points and it contains Gauss points that are closer to the element's boundaries (where the linear stress distribution attains a maxima). Consequently, the fully integrated element will predict localized failure initiation at a lower load level than the reduced integration element.

2. Global failure occurs at a lower load level with the fully integrated continuum element.

1 See Section 14.2 of The Finite Element Method by O.C. Zienkiewicz and R.L. Talyor, Fifth Edition, Butterworth Heinemann, Oxford, 2000.

- The difference in the global failure load predicted by the two elements is due primarily to difference in the local failure initiation load. The reduced integration element, with a single Gauss point per material ply, provides a more discretized representation of failure cascading. In other words, when failure occurs in a material ply of a reduced integration element, the stiffness of the entire ply is reduced and a relatively large amount of load re-distribution occurs. In contrast, when failure occurs at one of the Gauss points in a material ply of a fully integrated element, only part of the material ply experiences a stiffness reduction and a relatively small amount of load re-distribution occurs.

- One might be tempted to think that a fully integrated element inherently provides a more

realistic progressive failure response since it contains numerous integration points where the failure criteria is tested and stiffness reduction is imposed. This is not true because the integration points of a fully integrated element are not the most accurate locations for computing stress. In fact, the most accurate locations for computing stress are the reduced integration points1. Consequently, the reduced integration elements evaluate failure and stiffness reduction at fewer points, but the evaluation itself is more accurate because the stress state is more accurate at the reduced integration points. Therefore, the use of more integration points per element does not necessarily result in a more accurate analysis.

In comparing the failure response of the reduced integration continuum elements (C3D8R) with reduced integration continuum shell elements (SC8R), the following observations and explanations are given:

1. The continuum shell elements predict local failure initiation at lower load levels (42%) than the reduced integration continuum elements (52%).

- Even though both elements make use of the exact same set of Gaussian integration points, the SC8R elements predict higher in-plane stress components than the C3D8R elements. This discrepancy is due to differences in the transverse shear and transverse normal stiffnesses of the two elements. The C3D8R element obtains its transverse stiffnesses by simply integrating the stiffness of the individual material plies over the volume of the element. However, the SC8R element obtains its transverse stiffnesses as direct user input that applies to the element as a whole. This scenario effectively precludes the possibility of both elements exhibiting the same transverse stiffness. Any change in the transverse stiffness of an element will necessarily result in a different division of the element's total strain energy into in-plane and out-of-plane components. Therefore, if the two elements have different transverse stiffnesses, then they will likely have different in-plane stress components. In this particular problem, the initial matrix failure is primarily driven by the in-plane shear stress which is 19% larger in the SC8R element than in the C3D8R element; consequently, the SC8R element predicts earlier localized matrix failure.

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2. Global failure occurs at a higher load level with the continuum shell element even though local failure occur at much lower load levels.

- Even though the SC8R element predicts initiation of localized constituent failure at a lower load level than the C3D8R element, the SC8R element predicts that global structural failure occurs at a higher load level (68%) than the C3D8R element (60%). The fact that the SC8R predicts a more gradual failure cascade process is due primarily to the fact that localized material failures do not affect the transverse stiffnesses (E33, G13, G23) in the SC8R element (Abaqus requires these stiffnesses to be constant in the SC8R element). Since the transverse stiffnesses do not experience any degradation, the SC8R elements can more easily accommodate load re-distribution without causing additional localized failures. To summarize, the C3D8R and SC8R elements exhibit very different failure cascade behavior.

E1.4 Comparison of Helius:MCT with currently available Abaqus failure criteria Abaqus provides five different failure criteria that can be used to predict composite material failure in a linear elastic analysis. In addition Abaqus provides one type of damage evolution model that can be applied to composite materials. In this section, the various Abaqus failure criteria and damage evolution criteria will be compared with Helius:MCT by using each one to simulate the failure response of the composite conic structure.

E1.4.1 Comparison of Helius:MCT with linear elastic Abaqus failure criteria. In Abaqus Standard, five failure criteria are provided for use in linear elastic analyses: four stress-based criteria and one strain-based criterion. The practical utility of these linear elastic failure criteria is quite limited because they only predict the occurrence of localized failure, not the consequences of localized failure. In other words, when one of the Abaqus failure criteria predicts that failure occurs, there is no accompanying stiffness reduction. Therefore, the processes of load re-distribution and progressive failure are not represented. In contrast, Helius:MCT not only predicts localized failure, it also predicts localized stiffness reduction. In further contrast, the Abaqus linear elastic failure criterion utilizes the homogenized composite state of stress or strain to predict failure of the homogenized composite material, whereas Helius:MCT utilizes constituent average stress to independently predict failure of each constituent material. In this example problem, the composite conic can be assumed to be linearly elastic prior to global failure. In that respect, failure initiation predicted by Helius:MCT can be compared with the failure initiation predicted by the different Abaqus failure criteria. The failure criteria provided by Abaqus are applicable to generally orthotropic materials, but for the purpose of this example problem, a transversely isotropic material model is used for both the Abaqus failure criteria and Helius. One of the shortcomings of the Abaqus linear elastic failure criteria is that they are based entirely upon an assumed condition of plane stress; consequently they can only be used in plane stress 2-D continuum elements or shell elements. While Helius:MCT uses a general 3-D state of stress in its failure criteria, continuum shell elements (SC8R) are used in the comparison in order to accommodate the limitations of the Abaqus linear elastic failure criteria. In the case of composite laminates, laminate failure is taken to be first ply failure. For brevity, only the max-stress and Tsai-Wu failure criteria (stress-based failure criteria) are compared with Helius (stress-

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based). For more information on the max-stress and Tsai-Wu failure criteria formulation, see section 18.2.3 of the Abaqus Analysis User’s Manual. Model(s) SC8R.inp

• Failure criteria: Helius:MCT™ • Element type: SC8R • Through-thickness mesh density:


stressBasedFailure.inp

• Failure criteria: o Max-Stress and o Tsai-Wu

• Element type: SC8R • Through-thickness mesh density:


Results Table 5 shows the load level at which each type of failure event is predicted using the three different failure criteria. Figure 9 displays the load-vertical displacement curves for the load application point at 0°. Note that the max-stress and Tsai-Wu failure criteria only provide information on failure initiation for the homogenized composite material, hence these criteria predict that matrix failure and fiber failure both initiate at the same load level (50-51%). Helius:MCT predicts that the matrix constituent failure initiates at a load of 42% and fiber constituent failure initiates at a load of 43%.



Max Stress SC8R 2 4 51%4 51%4 n/a3

Tsai-Wu SC8R 2 4 50%4 50%4 n/a3

SC8R Helius:MCT™ SC8R 2 4 42% 43% 68%



3: Global failure is unable to be determined, see text for details

4: Composite laminate failure

stressBasedFailure

Model Element Type Matrix Failure1 Fiber Failure1 Global Failure1,2Criterion

Figure 9 displays the load-displacement curves for the load application point at 0°. Before global failure is achieved (68%), both methods produce similar predictions for overall structural stiffness. In using the linear elastic material failure criteria that are available in Abaqus, material stiffness degradation is not accounted for; therefore, the failure analysis can only proceed up to the point where localized failure initiation is predicted. The capability of Helius to degrade material stiffness at integration points that

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have failed (either matrix of fiber failure) lends the unique capability of accurately capturing the progression of local failure initiation into eventual global failure. Along with this, the user can accurately describe the post-fail behavior of the structure that is not available with the Abaqus failure criteria.


Failure initiation: 50% - Tsai-Wu criterion 51% - Max Stress criterion

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E1.4.2 Comparison of Helius:MCT with Abaqus’ progressive damage model For composite materials, Abaqus provides a progressive damage model that uses the Hashin criteria to predict the initiation of four different constituent failure modes and uses damage evolution equations to predict the post-initiation stiffness degradation that results from the evolution of each of the four constituent damage modes. This section provides a comparison between the progressive failure model used in Helius:MCT and the progressive damage model provided by Abaqus. There are several fundamental differences between the progressive failure model used in Helius:MCT and the progressive damage model provided by Abaqus. These differences are discussed below.

1. In the Abaqus progressive damage model, the material stiffness is gradually reduced as deformation continues to accumulate after the initiation criterion is met. In contrast, Helius:MCT imposes an instantaneous stiffness reduction that is determined by the particular constituent that failed. This type of instantaneous, discrete stiffness reduction normally poses severe convergence difficulties for finite element codes; however, Helius:MCT is specifically developed to efficiently handle this type of behavior and exhibits very robust convergence behavior.

2. In the Abaqus progressive damage model, damage initiation and damage evolution of the material constituents (fiber and matrix) are predicted based on the composite average states of stress and strain. In contrast, Helius:MCT uses the constituent average stress states to predict failure in each of the individual material constituents.

3. The Abaqus progressive damage model predicts damage initiation and damage evolution based solely on the in-plane stress and strain components, ignoring the contribution of the transverse stress and strain components. In contrast, Helius:MCT predict constituent failure using the full 3-D constituent average stress state. However, even if one ignored the contribution of the transverse stress components, the functional form of the in-plane stress contribution is still different in Helius:MCT and the Abaqus progressive damage model.

4. As damage evolves, the Abaqus progressive damage model only accounts for stiffness reduction in the in-plane stiffnesses (E11, E22, G12), leaving the transverse stiffnesses (E33, G13, G23) unchanged. In contrast, as material constituent failure occurs, Helius:MCT explicitly accounts for stiffness reduction in both the in-plane and transverse stiffnesses.

5. The Abaqus progressive damage model can only be used in conjunction with 2-D continuum elements and shell elements. Helius:MCT can be used in conjunction with 2-D continuum elements, shell elements, and 3-D continuum elements.

In this section, the Abaqus progressive damage model and Helius:MCT are each used to simulate the failure response of the composite conic structure. Due to the differences listed earlier in items 4 and 5 above, continuum shell elements (SC8R) are used in both models. This choice eliminates the differences listed in items 4 and 5; consequently, any differences in the predicted failure response of the structure are due entire to the differences listed in items 1-3 above. In addition to the fundamental mathematical differences listed above, the Abaqus progressive damage model poses addition difficulty to the user in that it is rather difficult and confusing to define. The process of defining the Abaqus damage initiation (Hashin) criteria is straightforward and requires only industry standard strength measurements (same as Helius:MCT):

• Longitudinal tensile and compressive lamina strength (+S11 and –S11) • Transverse tensile and compressive lamina strength (+S22 and –S22)

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• Longitudinal and transverse shear strength of the lamina (S12 and S23) However, the process of defining the Abaqus damage evolution relationships is difficult and confusing. The four parameters required for the damage evolution relationships represent the amount of energy that is dissipated in each of the four constituent failure modes:

• Energy dissipated during damage for fiber tension (Gcft)

• Energy dissipated during damage for fiber compression (Gcfc)

• Energy dissipated during damage for matrix tension (Gcmt)

• Energy dissipated during damage for matrix compression (Gcmc)

These energy dissipation constants are not readily available for most unidirectional composites (including AS4-3501-6). Furthermore, the convergence performance of the Abaqus finite element code is very sensitive to the numerical values chosen for these energy dissipation constants; therefore, even if the user has access to experimentally measured energy dissipation values, it is likely that the user will have to adjust these values in an effort to improve the convergence behavior of the finite element solution. In contrast, Helius:MCT does not require these energy dissipation values and Helius:MCT actually improves the convergence behavior of the finite element solution rather than degrading the convergence behavior. The progressive failure response of the composite conic structure was simulated using different models (listed below). Both models were identical except for the failure criteria that were employed. The SC8R.inp model used Helius:MCT to predict constituent material failure and stiffness reduction, and the Hashin.inp model used the Abaqus progressive damage model to predict constituent material failure and stiffness reduction. Model(s) SC8R.inp

• Failure criteria: Helius:MCT • Element type: SC8R • Through-thickness mesh density:


Hashin.inp

• Failure criteria: Hashin • Element type: SC8R • Through-thickness mesh density:


Results For this particular example problem, a converged solution could not be obtained using the Abaqus progressive damage model after the first localized fiber failure occurred. Consequently, the results reported here do not include a global structural failure load predicted with the Abaqus progressive damage model. Table 6 shows the load level at which each type of failure event is predicted using the two different progressive failure models.

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Hashin Hashin SC8R 2 4 50% 71% n/a3

SC8R Helius:MCT™ SC8R 2 4 42% 43% 68%


2: Global failure defined as large discontinuity in load-vertical displacement curve for 0° application point of load head

3: Global failure is unable to be determined, see text for details

Model Element Type Matrix Failure1 Fiber Failure1 Global Failure1,2Criterion

As seen in Table 6, the Hashin criterion predicts that localized matrix failure occurs at a higher load level (50%) than predicted by Helius:MCT (42%). This difference is due entirely to the differences in the matrix constituent failure criteria used by Helius:MCT and the Hashin model; e.g., constituent average stress versus composite average stress and the specific functional form of the stress-based failure criteria used in each model. Helius:MCT predicts that the first localized fiber constituent failure occurs at a load level of 43%. However, the Hashin criterion predicts that the first localized fiber constituent failure does not occur until the load level has reached 71%. The large difference in the load at fiber failure initiation is due primarily to two issues. First Helius:MCT uses the fiber average stress state, while the Hashin criterion uses the homogenized composite average stress state. Second, Helius:MCT predicts an instantaneous stiffness reduction in conjunction with localized matrix failure; therefore, load is re-distributed to the fibers much more rapidly than predicted by the Abaqus damage evolution model which uses a gradual stiffness reduction. Helius:MCT predicts that global structure failure occurs at a load level of 68%. Interestingly, the Abaqus damage evolution model did not even predict the initiation of localized fiber failure until a load level of 71%. It should be emphasized that after the initiation of localized fiber failure, a converged solution could no longer be obtained with the Abaqus damage evolution model, thus a global structural failure load could not be determined.

HeliusMCT v2 Example Problem 1 Abaqus

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