PROGRESSIVE PLY FAILURE ANALYSIS FOR COMPOSITE STRUCTURES

Barton W. McPheeters NEi Software, Inc.

5555 Garden Grove Blvd. Ste 300 Westminster, CA 92683

ABSTRACT

Design engineers working with composite materials typically use a linear Finite Element Analysis (FEA) solution and a failure index calculation based on the current state of stress in the model. However, this type of analysis can only provide accurate results up to first ply failure because of the linear assumption. This presentation will show how nonlinear progressive ply failure analysis can go beyond first ply failure and simulate subsequent damage propagation through a structure. This allows engineers to make a better assessment of conditions for ultimate failure so they can optimize their designs and also provide guidance on the most appropriate physical test program.

1. INTRODUCTION

In the last twenty years, composite materials, and in particular fiber reinforced composite plastic (FRP) materials have become widely used in many applications. Unfortunately, analytical techniques for designing with them have proven far more difficult than corresponding techniques for metallic materials. In recent years, a number of new techniques have immerged, including more realistic failure theories for dealing with the unique ways in which laminated composite materials behave and fail. This paper describes the implementation of one such analytical technique - - Progressive Ply Failure Analysis (PPFA), as implemented in the commercial Finite Element Analysis (FEA) software program NEi Nastran. The paper will attempt to demonstrate why progressive failure is an important consideration for the prediction of laminated composite behavior. The paper is intended for analysts who have used or are considering using FEA for the prediction of composite material behavior. It will not examine failure theories in detail except where they are important for understanding progressive ply failure. The paper will discuss the ways FRP materials are currently analyzed and the benefits of considering progressive failure when examining an FRP part.

2. BACKGROUND

While composites are valued for their cost, strength and weight, certain types have some less desirable characteristics that come with these benefits. These shortcomings include: brittleness, poor environmental resistance and poor temperature resistance. In addition, mechanical properties of composites are often dependent to a large degree on the manufacturing process.

From an analytical point of view, this variability in composites has presented problems. For any particular metal, Young‟s modulus (stiffness) is relatively constant, regardless of the variety or alloy. This is true of most commonly used metallic materials, and assists greatly in preliminary designs. On the other hand, this is not true of composites, and care must be taken to ensure that an appropriate material model is used, even in preliminary design. The additional variability of strength makes analytical determination of failure difficult as well.

Further, because of the brittle nature of many composites, yielding is not able to absorb energy from peak loading conditions. As a result, products are often vastly over designed in an attempt to account for the uncertainty. Metals, by contrast, in addition to relatively consistent and predictable properties, have a tremendous capability for energy absorption through plasticity.

However, in a laminated composite made of multiple layers of fiber plies, there is an equivalent mechanism that can be exploited. While individual plies may be brittle and fail catastrophically, a whole laminate usually does not. This is because while individual plies may fail, the entire laminate may not. Instead of a catastrophic failure reducing the strength of the laminate to zero, the load is often redistributed to other intact layers. This results in a “progressive” ply failure, meaning that after one ply fails, the strength is reduced somewhat, but not completely. Following this first ply failure, other plies may follow if the load is increased. In fact, even with many failed plies, a composite part may still have significant and useful load bearing capability.

Historically, most analytical assessments of layered composite structures were based on the concept of “First Ply Failure” (FPF). FPF means that the laminate is assumed to have failed when the first ply fails. Because this is a simple calculation to make, most analytical tools have the capability to assess failure based on FPF. While this is a good approach for situations where any degradation of strength is unacceptable, it provides no information about the residual load carrying capability of the “failed” part. In order to realistically determine design margins, it is useful to know if a laminate that has started to fail is going to come apart catastrophically, or if it is just going to crack a little and hold together. A progressive failure approach will provide this information.

Fortunately, today affordable computers have the capability to handle the added complexity needed for this type of analysis. As a result, PPFA is now within the realm of the typical composite engineer. This paper will demonstrate some of the potential uses of PPFA using NEi Nastran software, and illustrate the advantages of considering progressive failure in an analytical problem.

3. NUMERICAL APPROACH TO COMPOSITES

Most Finite Element programs use an approach to composites called Classical Lamination Theory (CLT). The CLT approach is linear, introduces no extra degrees of freedom, and is relatively straightforward. The properties of the individual plies are rotated into a uniform coordinate system and offset to represent their distance from the element‟s neutral axis. The stiffness contributions for all the individual plies are then added together. The element stiffness is then represented internally with a „smeared‟ stiffness that is the combination of all the individual plies. As noted, this approach does not require additional degrees of freedom, and can be easily used for linear analysis. Results at the individual plies are easily calculated from the element strains and curvatures from the smeared model. Reference [1] provides a good overview of the mechanics of this method.

Failure of a composite is different from traditional metals. In particular, metals are isotropic and have a limited number of failure modes -- tensile failure, shear failure, and also tend to be fairly ductile. In most cases, a metal will yield and slowly stretch until it fails, absorbing a significant amount of energy in the process. By contrast, fiber reinforced composites have many additional failure modes -- matrix cracking, bond shear failure, fiber/matrix separation, and individual plies tend to be quite brittle.

The von Mises yield criterion had been found to be a good analytical predictor of metal failure, but has been found lacking in the prediction of composite failure. This single number that indicates failure is a very useful way to make sense of a lot of analytical data. As such, a number of failure theories have been developed that attempt the same thing for composite materials. These theories attempt to produce a single-number that indicates whether a ply has failed. Some of these have been around for some time, such as the Hill and Hoffman theories [2]. Some have seen an increase in use lately, such as the Tsai-Wu theory [2]. A number of new theories have also appeared that attempt to match test results more closely. What most composite failure theories have in common is that like the von Mises yield theory, the stress state of the composite in different directions is combined to produce a single failure index. This is a single number that represents the failure status of the composite. If it is greater than 1, the entire ply failed. If it is smaller than one, it has not failed.

The Hill, Hoffmann, and Tsai-Wu failure theories [2] have been used successfully for years for many products. However, in reality, a ply failure in one direction or mechanism may not produce failure in another direction. For example, failure of the transverse matrix in a unidirectional tape changes the stiffness in the other direction relatively little. Some of the more modern failure theories, such as LaRC02 [3] and Puck [4] separate the different failure mechanisms to some degree. This approach allows a physically consistent numerical treatment of a partially failed ply.

In all cases, CLT can be easily modified to accommodate a reduced stiffness when an individual ply or portion of a ply fails. When the failure theory indicates that a ply has failed, it is a relatively simple matter to reconstruct the smeared properties, but with a modification to the stiffness contribution of the failed ply.

4. PROGRESSIVE PLY FAILURE ANALYTIC APPROACH

NEi Nastran, like most general purpose FEM codes, uses CLT for its composite material representation. However, NEi Nastran includes a couple of modifications to the basic approach. For example, CLT presumes that the plies are infinitely stiff in the out-of-plane direction, but NEi Nastran has a flexibility term built into its formulation to provide some degree of shear flexibility in the out-of-plane direction. This is an important modification, especially for relatively thick plies that may have significant flexibility in this direction or in thick laminates where the cumulative properties of many plies produce the same effect.

With CLT, any combination of plies can be combined to make up a laminate and represent it with the smeared properties. To run a progressive ply failure analysis, the status of each ply in a laminate is periodically checked against the chosen failure index. When the failure index indicates that a ply has failed, it is numerically simple to replace the representation of the intact ply with a failed ply, and then recalculate the smeared laminate properties. This is the approach that NEi Nastran takes.

For the traditional failure theories, failure is total once the failure index is greater than 1.0. NEi Nastran breaks this into two parts: failure of the ply, and failure of the bond. If the bond is shown to have failed, the intact bond stiffness is replaced with a reduced (or zero) stiffness. If the ply is shown to have failed, all of the ply properties (tensile stiffness in both directions and in-plane shear stiffness) are replaced with their failed stiffness.

The Puck [4] and LaRC02 [3] theories add a slight complication to this approach. Because failure indices are calculated for different failure mechanisms in these theories, an indicated failure may only apply to one different direction or property. In that case, only those directions or properties that are affected are changed. For example, if the transverse matrix is found to have cracked, only the E2 or transverse stiffness term is replaced. The primary stiffness E1 is left intact.

Because ply failure involves recalculation of the element stiffness, and subsequent recalculation of the model stiffness matrix, ply failure is necessarily a nonlinear problem. Therefore, NEi Nastran only has this option available in nonlinear static and nonlinear transient solutions. To save computation time, NEi Nastran only updates the stiffness and checks for ply failure at the end of a converged increment. Ply failures are not calculated as part of the intra-increment equilibrium iterations. The advantage of this approach is that it avoids the nonlinearity of many plies potentially failing during iteration and producing an unstable situation. The disadvantage is that large increments may produce somewhat discontinuous results, as all the plies that fail are made to fail together at the start of each increment, rather than continuously during the increment. Fortunately this situation is relatively simple to identify and correct by breaking the loading up into more increments. The saving is so significant that it is still faster to make multiple runs to find a continuous solution.

5. ADVANTAGES OF PROGRESSIVE PLY FAILURE ANALYSIS

The ability to analyze partial failure gives engineering analysts a tool to assess the load carrying capability of a partially failed composite laminate. This allows several questions to be answered:

Is the failure of the first ply associated with abrupt failure of the component?

In some cases, the load may be such that the residual load carrying capacity of a part is insufficient to maintain integrity with a damaged laminate. In that case, a progressive ply failure analysis will show a swift failure of all remaining plies once the first ply has failed. This is useful information for determining margins of safety to incorporate into a design.

Is the stress re-distributed over adjacent plies maintaining the load-carrying capability of the part?

In some cases, failure of a single ply will simply result in the load being taken by other plies in the laminate. In this case, a progressive ply failure analysis will show that equilibrium is reached after failure of some plies, but that part does not fail catastrophically. Like the abrupt failure case, this information can assist in determining design margins.

Is load re-distributed over adjacent elements?

It is possible that the partial failure of a section will redistribute the load to other parts of the structure. A progressive failure analysis would show where the loads end up, and can provide valuable insight into the design of the other members in order for them to take the additional load properly. An option may exist to channel the redistributed load away from critical members into ones designed for the additional load.

How far is First Ply Failure from Last Ply Failure?

If the last ply fails relatively quickly after the first ply, it may make sense to use a FPF based design approach for the rest of the model, or for the rest of the design cycle. However, this would be hard to verify without at least one progressive failure analysis.

What is the evolution of the part from FPF to LPF failure?

Because of the nonlinear nature of a progressive failure analysis, it should be noted that the progression from first ply failure to last ply failure is not necessarily a straight line. In fact, it maybe that a few plies fail initially, but that the remaining plies remain intact for a large portion of the loading cycle, only failing catastrophically at the very limit of the load. This information maybe able to provide the basis for a redesign that eliminates the early failure, thus extending the range of the intact laminate considerably. Similarly, if most of the plies fail

immediately, and the last ply only fails at the very end, it implies that the initial damage takes away most of the load carrying capacity of the laminate. In this case, a FPF based approach would also work reasonably well.

6. PROGRESSIVE PLY FAILURE EXAMPLES

This paper will examine several small models that demonstrate how partial failure is analyzed and what kinds of results can be expected from such an analysis. For all cases except the wing, a multilayer generic woven fiberglass-epoxy composite with 16 layers is used. The following examples will be presented and discussed.

A simple small cantilever plate subject to a bending load. This model will illustrate the basic concepts embodied in a progressive failure analysis.

A simple plate with a hole subject to a tensile loading. This model demonstrates several different failure mechanisms and shows how loads redistribute.

A box beam subjected to an enforced end deflection. This model will show the evolution of a failure and why that is important.

A simple impact problem where the bending load of the impact locally fails some of the plies in the laminate.

A realistic wing model that illustrates an abrupt failure.

6.1 Small Cantilever Demonstration Model

Figure 1 shows this model. It is a 10 element plate that is loaded with an end load. The end load is applied in three loading cycles. The first cycle is a low load, and the beam bends slightly, and a few elements fail. The next loading cycle is substantially higher than the first, and many of the plies in the model fail. The third cycle is the same as the first, but now it is applied to the failed model.

In the first loading cycle, a number of the extreme plies fail, but the majority of the model remains intact. As a result, the model is largely intact after the first cycle. Figure 2 shows the damage following the initial loading cycle. In the second cycle, most of the plies in the first element fail, and the stiffness is reduced considerably. Figure 3 shows the model after the second loading cycle. Note the almost total failure.

Finally, the initial loading is repeated. The second time, however, the model has a considerably reduced stiffness as a result of the failed plies. As a result the third cycle shows a tip displacement almost as large as the large load in the second cycle. Figure 4 is a graph of tip displacement as the model is loaded and unloaded.

This illustrates in a nutshell the basic concept of a progressive failure analysis. When the model is pushed beyond its elastic limits, some or all of the individual plies in the model fail, and result in a model that is much more flexible. Deflections in future loads will be greater than those expected in an intact model.

Figure 1. Demonstration plate model.

Fixed End

Upward Load

Figure 2. Composite failure status after initial loading cycle.

Figure 3. Composite failure status after three loading cycles.

Minor Failures

No Failures

% of plies failed

Almost total failure – severe stiffness degradation

Still no failures

% of plies failed

Figure 4. Tip deflection of plate through three loading cycles.

6.2 Simple Plate Model

Figure 5 shows a simple plate model with a hole in it. The plate is 10 units long, 6 units wide and with a 2 unit hole in the center. A load sufficient to fail the plate is applied in 20 increments, placing the part in tension. As the load is ramped up, elements and plies around the hole start to fail. By the time the full load is applied, most of the plies in the area of the hole have failed, as well as most of the elements across the section. Once all of the plies have failed, the part has significantly less stiffness in the loaded direction, and the nonlinear loading iteration becomes more difficult. A plot of the end displacement vs. load shows that the stiffness is roughly constant until all the plies have failed, at which point the stiffness decreases quickly and dramatically.

This model illustrates a number of things unique to a progressive ply failure analysis that would not show up in a linear first ply failure analysis.

The initial (first ply) failures are in the elements around the hole and show up in the fourth load increment, corresponding to a load of roughly 30% of the ultimate failure load (Figure 6). This is not surprising. However, closer examination of the failed portions shows that the elements at the end are placed in slight compression. Because the compressive strength of this material is very low (compared to their tensile strength), even a slight compressive load causes failure. The elements along the side of the hole are failing in shear, as the shear strength of this material is also very low. However, the load carrying capacity of the plate, which is provided almost entirely by tensile loading of the fibers, is practically unchanged. If this plate were subject to a first ply failure analysis, the load carrying capacity of the plate would be listed at 30% of the ultimate load. The stiffness of the plate is relatively unaffected up to a load of almost 85% of the ultimate load (Figure 7). At that point, the fibers aligned with the load begin to fail and

Peak of first cycle

Peak of second cycle

Peak of final cycle

Minor Ply Failures

Severe Ply Failures

Linear ramp

No further failures

Time (sec)

Tip Deflection (normalized)

the stiffness begins to degrade quickly. At 100% of the ultimate failure load (Figure 8) the last of the fibers aligned with the load fail in tension and resistance to the applied load disappears almost entirely. Figure 9 illustrates the end deflection of the plate as the load is increased. Despite the early failures of some plies, this part has significant reserve load carrying capacity.

In a situation like this, it might be possible to redesign the plate or laminate in some way to minimize the compressive and shear loading around the hole. It is probable that a design could be found that survives mostly intact. In fact, with the information from this analysis, it is probable that a new arrangement of plies could be found that carries more load and postpones first ply failure with the same amount of material and number of plies. This model illustrates a situation where first ply failure and last ply failure are separated by almost an order of magnitude in loading.

In this particular model, the failure stiffness ratio (failed stiffness/intact stiffness) is set to 0.04 (or 4%) in the tensile directions and 0.20 (20%) in the shear directions. This means that once a ply fails in tension or compression, the stiffness is reduced to 4.0 percent of the intact stiffness. You can see the effect of this in the model by looking at end deflection versus load. As the load increases and more plies fail, the deflection gets greater for each incremental increase in load. However, because the tensile plies are not failing until late in the analysis, this effect is small until failure is almost complete. At the end, in the „completely failed‟ state, the plate still retains roughly 4% of its original stiffness, and the stiffness curve shows this as a change in slope.

Figure 5. Simple plate model.

Fixed Edge Uniform Load

Figure 6. Simple plate model at 30% of ultimate load.

Figure 7. Simple plate model at 85% of ultimate load.

Initial Shear Failures

Initial Compressive Failures

% of plies failed

Area of totally failed elements

Some plies have failed, but laminate is still

largely intact in the load direction

% of plies failed

Area of total failure has almost reached completely across

Figure 8. Simple plate model at ultimate load.

Figure 9. Plot of end deflection vs. load.

Total failure has occurred across the entire section

% of plies failed

Load Increment (.65=ultimate load)

Initial Ply Failures

Tensile Fibers Begin to Fail

Deflection w/o Failure

Area of Some Stiffness Degradation

Final Failure

End Deflection (normalized)

6.3 Box Beam

This model illustrates the evolution of a failure. In this model, shown in Figure 10, a box beam is loaded with an enforced tip displacement. In order to illustrate the failure better, only a half model is used with symmetry boundary conditions. Further, in order to promote failure away from the boundary conditions, the beam is constructed of two composite sections. The part close to the boundary is a thick (16-ply) laminate, while the rest of the beam is an 8-ply layup.

As the tip of the model is displaced, the top of the beam is placed in tension and the bottom section in compression. The material used in this model is much stronger in tension than compression, so the first failures appear on the lower section of the beam, as in Figure 11. As the lower section fails, the load is transferred from the lower section to the side walls of the beam. However, the side walls are not any stronger than the bottom, and they begin to fail quickly as well, from the bottom up. Figure 12 shows the composite failure status at the point where the stiffness of the side and bottom sections has been almost completely lost.

As the deflection is increased, the failure of the side walls becomes complete, leaving only the top section of the beam intact. At that point, the bottom portion of the beam begins to buckle, resulting in another decrease in the applied force. Figure 13 shows the deformed state and the composite failure status as the failed sections begin to buckle.

The graph in Figure 14 shows the applied force necessary to enforce the deflection. Note the points on the graph where the bottom begins to fail, and quickly completes failure. As the side walls fail, the force necessary to continue the displacement decreases dramatically until only the top portion is left intact. For a time this holds, with the failed bottom and side walls maintaining their shape and resisting the increased load. Eventually, however, the bottom buckles and the failed plies no longer provide any resistance to the applied load. At that point, the load is reduced once again. Further deflection of this beam can be done with almost no additional force, as the only resistance is provided by the top section in bending.

Figure 10. Box beam model.

Figure 11. Initial failure on bottom section of box beam.

Rigid Spider

to Input Point

Input Point

(Enforced Downward Displacement)

Symmetry boundary conditions

Fixed End

Thick Composite

Thin Composite

Initial Failures on Lower Surface

(At boundary with thick and thin composite)

% of plies failed

Figure 12. Point of catastrophic stiffness degradation.

Figure 13. Buckling failure point.

Failure almost complete

through vertical section

% of plies failed

Vertical section failure is complete

Bottom surface begins to buckle

Top surface is only load

carrying area left

% of plies failed

Figure 14. Applied force needed to deflect box beam.

6.4 Impact Model

Impact is generally considered an event best modeled with an explicit finite element code, as it usually takes place quickly and produces significant nonlinear effects. However, for a relatively slow impact, where most of the energy is absorbed in bending, an implicit code may be appropriate.

Impact on composites almost always results in local failures, but a progressive failure analysis will tell us if the failures are important. In this example, a ball projectile locally fails the extreme fibers of the plate, but the internal elements are intact. As such, the load carrying capacity of the plate is almost intact, with some local load redistribution to the inner plies and to elements around the impact site.

Bottom surface begins to fail

Failure spreads through vertical section

Total failure of bottom and vertical sections

Bottom buckles

Linear Elastic Loading

Load increment

Force at tip (normalized)

In this example, we have a slowly moving ball that impacts a pinned-edge plate. Figure 15 shows the model. The plate bends, absorbing the impact, and then rebounds, throwing the ball back upward. In the course of the impact, the area of failed plies increases as the ball deflects the plate. However, only a small portion of the plate undergoes a complete failure. In most of the impact area, only slightly more than half the plies have failed. Figure 16 shows the extent of the failed plies, noting that outside the impact zone, the laminate in most cases only has a few failed plies. In most of these areas, the failures are in the weak directions of the individual plies. That is, plies that are not oriented to follow the load fail in their weak directions. In addition, a fairly large area of high shear stress develops, and because shear is also a weak direction, we see failures in heavily sheared elements. However, and this is important for the problem, there are a limited number of tensile failures, and thus the stiffness of the plate is relatively unchanged except directly under the impact.

It is these failures, however, that use up energy. When the plies fail, the plate in the impact zone becomes softer and lower in frequency, imparting less energy to the rebounding ball. Figure 17 shows a graph of the ball velocity as it hits the plate and rebounds. On the same graph is a plot of the ball velocity when the plies were not allowed to fail. Note that the rebound velocity of the plate that has failed is lower, and that the ball separates from the plate somewhat later due to the lower frequency.

In this case, a progressive failure analysis had told us a number of things. First, for this impact, there is limited catastrophic failure. Second, because most of the failures are in shear or in the off-direction, we do not see a significant reduction in stiffness for this plate, despite some total failures in the impact area. And finally, we saw how ply failures can absorb energy like plasticity in metals.

Figure 15. Impact model.

Pinned Edges

Initial Velocity

Pinned Edges

Composite Plate

Figure 16. Impact model showing maximum extent of failed plies.

Figure 17. Ball velocity for progressive failure model and linear model.

% of plies failed

Almost total failure in this area

No failed plies in this area

Upward velocity

following rebound

Rebound velocity with failures

Rebound velocity w/o failures

Ball leaves plate

Time (sec)

Ball Velocity (normalized)

6.5 Large Model

This model is more typical of a model that might be used for a progressive ply failure analysis. In this analysis, a composite wing structure is loaded with a simple end load. Because of the complex internal structure, the failure of the various parts is considerably more complicated than the previous two examples. Figure 18 shows the model used for this example. The laminate in this case is a mixture of unidirectional and woven plies with different layups in different places. The woven and unidirectional materials are generic fiberglass/epoxy composites.

As it turns out, the primary failure point is, as is often the case, at the bolted attachment. In this case, a number of plies in different directions fail in relatively quick succession around the bolt. The space from first ply failure to last ply failure is relatively small, and the damage is confined to a relatively local area. Figures 19 and 20 show progressive damage as the plies first begin to fail around the bolts, then almost completely fail by the next load increment. If the analysis continues, the failed area increases and the bolted supports continue to rip out a large portion of the lower surface, as shown in Figure 21.

Unlike the first example, there is relatively little reserve capacity in this model. Once failure starts, it continues to total failure quickly. However, the progressive analysis has allowed us to make that call. In the absence of the analysis, the reserve capacity would be unknown, and a little local failure around the bolts might have been dismissed as inconsequential to the system. The analysis has shown, however, that the failure, while local, would severely impact the operation of this part, as its attachment is insufficient.

In this model, it is necessary to redesign the bolted connection area, possibly adding more material, or adding backing plates to spread the bolt loads around somewhat.

Figure 18. Large model.

Figure 19. Large model showing initial failed plies on under side bolts area at 20% load.

Load Point

Symmetry Plane

Bolted Support Points

First plies failing around bolt on underside

% of plies failed

Figure 20. Large model showing almost total failure at 30% load.

Figure 21. Large model showing regions of failure at 100% load.

Plies beginning to fail here

Laminates here have progressed to almost total failure

% of plies failed

Area of failure at 100% of load

Areas of total laminate failure % of plies failed

7. CONCLUSIONS

There are several conclusions to be drawn from these examples. First, is it important to know the failure mechanism and reserve load capacity of a part before you can intelligently assign design factors or margins of safety. A progressive ply failure analysis provides a lot of information that would not be available from a simple linear analysis using a first ply failure technique. Even for parts that are not expected to fail, even partially, a progressive ply failure analysis can provide valuable information. An analysis of this type will provide information about such items as: reserve load capacity, potential for catastrophic failure, and load redistribution.

8. REFERENCES

1. Jones, Robert M., Mechanics of Composite Materials, New York, Taylor & Francis, Inc., 1999.

2. “Laminates,” NX Nastran User’s Guide, Chapter 22, UGS Corp., 2007.

3. Dávila, C.G., Jaunky, N., and Goswami, S. “Failure Criteria for FRP Laminates in Plane Stress.” 44th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics, and Materials Conference, Norfolk, Virginia, 7-10 April 2003.

4. Puck, A. and Schürmann, H. “Failure analysis of FRP laminates by means of physically based phenomenological models.” Composites Science and Technology,” 62:12-13 (2002) 1633-1662.