Computerized calculation of composite laminates and structures: theory and reality

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1 Computerized calculation of composite laminates and structures: theory and reality M. Sonnen, C. Laval, A. Seifert 2004 MATERIAL S.A. presented at: SAMPE 2004, May 19, 2004, Long Beach, California, USA Abstract The database and design software for composites, which has been developed by the authors, is now being used in research and industry. The reactions from composite design experts worldwide using the software give new insight into the calculation of composite laminates and structures. The differences between the major commonly used and new failure theories as well as progressive failure analysis and degradation models are explained. They are compared with the help of two typical real world applications and case studies (a pressure vessel and a tube under torsional load). This provides an overview of the features, properties and qualities of the different theories and models. In which cases the results are similar, when do they differ and how important these differences are will be discussed. 1. Introduction The basics of the laminate calculation, the Classical Laminate Theory (CLT), is widely known and accepted [1, 2]. The CLT allows the calculation of the laminate's properties as well as the stress and strain distribution for each ply in the laminate. Once the stresses in the plies are known the strength of the laminate can be calculated by applying a failure criterion to each ply. The strength of the weakest ply is the first ply failure strength of the laminate, which in general is not its ultimate strength. The laminate's ultimate strength can be predicted by progressive failure analysis after the first ply failure. A large variety of failure criteria and degradation models have been developed over the years. They cause ongoing discussions about the correctness and capacity of the different theories. The database and design software for composites developed by the authors allows the user to compare these theories on any laminate-load-case very quickly. Two typical case studies calculated with the software and compared with real world tests show the differences and similarities of these calculation concepts.

2 2. The software - a brief overview The software developed by the authors is an all-embracing database and design software and for composite materials and structures. It calculates the properties of plies by micromechanics the properties of laminates by Classical Laminate Theory (CLT) or Netting Analysis the laminate load response including the consideration of hygrothermal loads or other preloads failure with global failure criteria (Hoffmann, Tsai-Hill, Tsai-Wu) and differentiating failure criteria (Simple Puck, Modified Puck, Hashin, Puck Action Plane) and progressive failure models (Tsai, Puck) the properties, load response and failure of basic composite structures (e.g. beams, tubes, plates) Besides the above mentioned calculation capabilities the following substantial features have been integrated in the software: FEA interface: Complex composite structures have to be analyzed by Finite Element Analysis (FEA) programs. There is a wide range of FEA software available. Some are more suitable for composite materials than others. To be able to use state-of-the-art laminate calculation together with existing FEA programs an extended FEA interface has been implemented. Database: To use the calculation capabilities of the software efficiently a powerful database is needed. This database stores all data of fibers, matrices, plies, laminates, stacking sequences, loads and structures. Features like searching, filtering and sorting allow direct access to the data for immediate calculation. In addition to physical and engineering parameters information like comments and text notes or quality control data can be stored. Graph engine: One of the best way to present data is a graph. The software includes a graph engine, which allows the display of any X-Y or polar graph, including any ply's property versus the ply's orientation angle, carpet plots, ply-by-ply stress and strain diagrams and failure process diagrams. An important tool is the display of the 3D failure envelope surface in the (s 1,s 2,t 12 ) stress space (Fig. 1). User interface: Software is only as good as its user interface. Therefore great importance has been given to the user interface. The aim was to make the software as efficient as possible. The spreadsheet style use makes it possible to change parameters and calculation theories in a fraction of a second.

3 Fig. 1: 3D model of the Puck Action Plane Criterion failure envelope with constant and variable stress vector 3. Failure criteria To calculate the strength of a laminate the stress state in each ply has to be submitted to a failure criterion. The first criteria developed for unidirectional plies were global failure criteria. Global criteria do not distinguish between failure modes. They were formulated as a single mathematically simple equation (the calculation power of modern computers was not available yet) which are easily adaptable to the test results. They are generally not based upon the physics of the failure mechanisms. Because of the quadratic form of the equation these criteria are often referred to quadratic criteria. Examples are: Hoffmann [3], Tsai-Hill [4] and Tsai-Wu [4]. Until today one of the most commonly used criterion is the Tsai-Wu criterion. Physical observations led to the development of differentiating criteria, which distinguish between fiber failure (FF) and inter fiber failure (IFF). Different mathematical formulations are used for the physically different phenomena FF and IFF. Because the effects of the two failure modes and the methods to avoid them are completely different, it is vital for the designer to know which failure is occurring. Examples for such criteria are: Simple Puck [5], Modified Puck [6] and Hashin [7]. Based on physical models Puck developed one of the most modern differentiating criterion in order to integrate the numerous experimental observations into one theory [8]. This Puck Action Plane Criterion does not only distinguish between FF and IFF but also between three different modes of IFF (Fig. 2, 3). Especially the phenomenon of oblique IFF with s 2 < 0 (Fig. 3: mode C) motivated Puck to identify the ply plane with the highest effort as "action plane" and to transform the fracture calculations for brittle materials into that plane. Whereas the IFF modes A and B are often tolerable, mode C can lead to a catastrophic failure of the complete composite part. If the angle of the fracture plane exceeds ±30 the wedge shape of the crack then can damage the neighboring plies in the laminate and thus result in an explosive failure of the whole laminate.

$Fig. 2: fiber fracture modes Fig. 3: inter fiber fracture modes 4.$ After a ply in the laminate reaches its IFF limit, local matrix cracks will appear. These cracks do not lead to the total loss of this ply's strength since neighboring laminate bridge them.

After a ply in the laminate reaches its IFF limit, local matrix cracks will appear. These cracks do not lead to the total loss of this ply's strength since neighboring laminate bridge them.

4 Fig. 2: fiber fracture modes Fig. 3: inter fiber fracture modes 4. Degradation models and progressive failure analysis Together with degradation models the progressive failure analysis is able to simulate the laminate's failure process after the first ply failure. After a ply in the laminate reaches its IFF limit, local matrix cracks will appear. These cracks do not lead to the total loss of this ply's strength since neighboring laminate bridge them. However there is a reduction in the ply's stiffnesses. This results in a new distribution of the stresses in the laminate. With increasing load beyond the IFF limit more and more local matrix cracks will appear until a certain saturation is reached [8, 9]. In laminate calculation the physical reduction of the stiffnesses is taken into account by degrading the ply i.e. by mathematically decreasing the ply's stiffnesses. In the respective literature only a few degradation models can be found. Mostly the plies are degraded by a constant factor directly after the IFF [2, 4, 10]. If a global criterion is used it is impossible to distinguish between IFF and FF. Tsai therefore suggested that the first failure of a ply is always considered as IFF and the related properties E 2, G 12 and n 12 are degraded by a constant factor F IFF. The second failure of the same ply is considered as FF and the properties E 1, E 2, G 12 and n 12 are degraded by a constant factor F FF. This does not reflect the crack building process as described above. For that reason Puck introduced a new degradation model together with his Action Plane Criterion [8]. The degradation of the ply follows a non-linear function of the effort. In order to simulate this process on the computer an iterative algorithm is used where the load is increased step by step. After each step the complete laminate as well as the stress distribution has to be recalculated and over stressed plies have to be degraded. Normally a fiber failure in the laminate is not acceptable and the progressive failure analysis is stopped. However

5 laminates with several fiber orientations can still have load-bearing capacities. Consequently it can be interesting to continue the calculation beyond the first fiber failure until the last ply fail in fiber direction. 5. Theories compared To show the similarities and differences of the different theories following are two case studies of typical composite parts Case study 1: a typical filament wound pressure vessel The first example examines a typical 250 bar filament wound pressure vessel as produced industrially by one of the authors' customers. The main dimensions are: inner laminate diameter: 380 mm ply material: carbon fiber T700 / epoxy 60% working pressure: 250 bar burst pressure: 2.5 * working pressure = 625 bar Fig. 4 shows the vessel's laminate original structure (Vessel 1) as well as a modified version without the +/- 55 plies (Vessel 2). Vessel 1 Vessel 2 Ply Angle [ ] Thickness [mm] Ply Angle [ ] Thickness [mm]

6 Fig. 4: laminate structure of the pressure vessel Total: Total: The stresses in the cylindrical section of a pressure vessel under internal pressure load are characterized by bidirectional stresses with a ratio radial:axial = 2:1. All stresses are positive. Basically the +/- 88 plies bear the radial stresses, the +/- 11 plies bear the axial stresses. The +/- 55 (ply 7 and 8) can often be found in the stacking sequence of filament wound tubes to withstand internal pressure. There are resulting from the Netting Analysis: ~ 55 is the resulting angle for the axial and radial forces. Because of its low stiffness the thermoplastic liner is not taken into account. In addition the inflating effects of the complete vessel are ignored. For the analysis three criterion-degradation-combinations are used: Puck's Action Plane Criterion with Puck's degradation model Puck's Action Plane Criterion with Tsai's degradation model Tsai-Wu failure criterion with Tsai's degradation model Fig. 5 shows the results. There are no significant differences between the calculation methods. The first reason for that is that the failure envelops of the Tsai-Wu and the Puck's Action Plan Criterion are very similar for positive stresses transverse to the fiber direction (s 2 >0). However for load cases with s 2 <0 the differences between the methods would be more important. Vessel 1 Vessel 2 Failure criterion: Puck Puck Tsai/Wu Degradation: Puck Layer Tsai Layer Tsai Layer Load factor at first IFF: Load factor at last IFF: Load factor at first FF: Num. of failure events: Load factor at first IFF: Load factor at last IFF: Load factor at first FF: Num. of failure events: all +/ all +/ all +/ all +/ all +/ all +/ all +/ all +/ all +/ all +/ all +/ all +/ all +/ all +/ all +/ all +/ all +/ all +/- 88 Load factor 1 º 250 bar working pressure Fig. 5: calculation results for the pressure vessel The second reason is that the influence of the degradation in this example is very small. In this kind of laminate with two fiber directions almost perpendicular to each other the fiber's stiffness transverse to its fiber direction E 2 is not important. After an inter fiber failure the degradation only effects the Young's modulus E 2 (Fig. 6 shows the degradation factor for E 2 for different plies), the shear modulus G 12 and the Poisson's ratio n 12 but not the modulus in

7 fiber direction E 1. To verify this we calculated the laminate with and without degradation (degradation factors = 1). The calculation results in Fig. 7 confirm that the influence of the degradation can be ignored. Fig. 6: degradation factors for E2 according to Puck for the plies 1 (88 ), 7 (55 ) and 9 (11 ) Vessel 1 Failure criterion: Puck Puck Degradation: Puck Layer none Layer Load factor at first IFF: Load factor at last IFF: all +/ all +/-88 Load factor at first FF: all +/- 88 Num. of failure events all +/ all +/ all +/- 88 Fig. 7: results with and without consideration of the degradation Finally we would like to discuss the influence of the "traditionally" used +/- 55 plies. The calculation results in Fig. 5 shows that Vessel 1 is a little bit stronger that Vessel 2. However the wall thickness of Vessel 2 is 6% less. The influence of the +/- 55 originates only from its thickness not from its fiber orientation.

8 Fig. 8 shows the pressure vessel after total failure. Fig. 8: pressure vessel after total failure This example of a pressure vessel shows: the differences between the failure criteria are not important the degradation of inter fiber failed plies is not important the "traditionally" used +/- 55 plies are not needed 5.2. Case study 2: a tube under torsional load A tube under torsional load is a very good example to demonstrate different kinds of failure modes. Tensile and compressive forces are at work. The optimum fiber orientation to bear a torsional load is +/- 45. The basic dimensions are: inner laminate diameter: 50 mm laminate structure: 2 plies (+45 and -45 ), fiber glass / epoxy 60% ply thickness: 1.5 mm torque moment: 50 Nm Fig. 9 shows the failure process results calculated with the Tsai-Wu failure criterion and the Tsai degradation model. The Tsai-Wu criterion does not distinguish between different failure modes. As described in section 4 Tsai considers the first failure in a ply as inter fiber failure (IFF) and the second failure as fiber failure (FF). Thus as Fig. 9 shows the -45 ply has a IFF at load factor and a FF at load factor Fig. 9: failure process for the tube under torsional load (criterion: Tsai-Wu, degradation: Tsai)

Now we would like to apply Puck's Action Plane Criterion with Tsai's degradation model. Fig. 10 shows the results. They are very close to the first results.

9 Now we would like to apply Puck's Action Plane Criterion with Tsai's degradation model. Fig. 10 shows the results. They are very close to the first results. However the differences between the failure criteria are noticeable. Fig. 10: failure process for the tube under torsional load (criterion: Puck Action Plane, degradation: Tsai) Now we would like to use the Puck Action Plane Criterion with the Puck degradation model. As Fig. 11 shows the results differ significantly from Tsai's theory. The first failure is a tensile IFF (Puck mode A) in inner ply at load factor followed by a shear IFF (Puck mode C) in the outer ply. Because the angle of the fracture plane is >30 the resulting wedge will destroy the complete laminate. This failure is critical. The compressive FF in the inner ply at load factor will never occur. Fig. 11: failure process for the tube under torsional load (criterion: Puck Action Plane, degradation: Puck) When manufacturing such a torsional tube and testing it the failure modes as predicted by Puck's theory, tensile FF in the inner ply and shear IFF in the outer ply including the wedge effect, can easily be recognized (Fig. 12). Fig. 12: shear IFF of the tube under torsional load with wedge shaped fracture This example of the tube under torsional load shows:

10 the differences between the failure criteria are significant the different degradation models predict completely different results tests show that in this case Puck's theory describes the failure process correctly 6. What's in it for the real world? The examples above show that the description of some cases with the help of global criteria is not detailed enough. Here Puck's theory delivers more precise results and provides the designing engineer with a better insight into the failure process. However, it seems to be important, to have the possibility of comparing a well established, well know theory with new, more detailed calculation methods. The reflections about the results will provide a better understanding of the matter and give a greater reinsurance of their correctness. The reactions from composite design experts worldwide using the software confirm this. The development of calculation theories will continue. Almost certain new phenomena will come up which can not be described by the existing theories. Therefore the authors will continue to follow the latest developments and will constantly integrate new theories into their software. 7. References 1. C. Herakovich, Mechanics of fibrous composites, John Wiley & Sons, Inc., New York, J.-M. Berthelot, Composite materials: mechanical behavior and structural analysis, Springer, New York, O. Hoffmann, The brittle strength of orthotropic materials, Journal of Composite Materials, 1, April 1967, pp S. Tsai, Theory of composites design, Think Composites, Dayton, A. Puck, W. Schneider, On failure mechanisms and failure criteria of filament-wound glass-fiber/resin composites, Plastics and Polymers, February 1969, pp A. Puck, H. Schürmann, Die Zug/Druck-Torsionsprüfung an rohrförmigen Probekörpern, Kunstoffe, 72, 9, 1982, pp Z. Hashin, Failure criteria for unidirectional fiber composites, Journal of applied mechanics, 47, June 1980, pp A. Puck, Festigkeitsanalyse von Faser-Matrix-Laminaten, Carl Hanser Verlag, München, N.N., Development of FRP components (fiber reinforced plastics), Calculation, VDI guideline 2014, part 3, Verein Deutscher Ingenieure, Düsseldorf, in press 10. R. Jones, Mechanics of composite materials, second edition, Taylor & Francis, Philadelphia, BIOGRAPHIES Axel Seifert (mechanical engineer) and Christian Laval (electrical engineer) worked from 1986 to 1990 in the composite department of the IKV (Institute for Plastics Processing) at the Technical University Aachen, Germany. In 1990 they founded MATERIAL S.A., a company active in the field of software, design, engineering and prototyping

11 of composite materials. Michael Sonnen (mechanical engineer) worked from 1989 to 1993 in the composite department of the IKV. From 1993 to 1997 he was the head of the composite group of Verseidag Indutex. Since 1997 he has been responsible for engineering services at MATERIAL S.A.

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