An improved strength tensor-based failure criterion with stress interactions
In the past decades, various failure criteria models have been developed and modified for industry to predict the strength of parts. Some models were designed for unidirectional fiber-reinforced composites and cannot be extended to model failure in systems with fiber orientation distributions or to textile-reinforced composite structures. In fact, none of the existing models account for all modes of failure,1, 2 nor do they include all stress interactions, if any. Some only include interactions between normal stresses, such as the interaction between longitudinal and transverse normal stresses.3–7 Others only include interactions between transverse shear stress and transverse normal stress.8–14
We have developed a tensor-based model incorporating fourth-order tensor components that represent the interactions between stresses. Consequently, it can be applied to any anisotropic material, such as continuous, chopped, and woven-fiber-reinforced systems subjected to complex stress fields, as depicted schematically in Figure 1. This model does not consider phenomenological aspects of failure, such as type and direction during failure, but rather is a purely mathematical model that will be easily implemented in industry to predict failure.
The failure criterion presented here15 is a strength tensor model based on the Gol'denblat-Kopnov criterion.4 The model defines a scalar failure function, f, as a function of strength tensors and stresses. For example, for a plane stress case the criterion can be written as2. Hence, minimal experimental data is needed to compute the slope of the failure surface around any of the four strength values within the normal stress failure plane, such as the σ11−σ22 plane. To compute F1122, one may choose any of the following equations:
As Figure 2 shows, for the normal stress–shear stress failure plane, such as the σ22−τ12 plane, one slope can be used and the interaction term is given by15 for the set of strength tensor components required with the Gol'denblat-Kopnov and the Malmeister or Tsai-Wu models.
We tested the model by comparing it with other models and experimental data for unidirectional fiber-reinforced polymer materials, as well as with experimental data for an anisotropic paperboard and a two-ply plain weave glass-fiber-reinforced polymer laminate. For example, we analyzed the failure of unidirectional composites in the σ11−σ22 and σ22−τ12 stress planes using our model, and compared it with first World Wide Failure Exercise (WWFE-I) experiments14performed with glass-fiber-reinforced epoxy composites. Figure 3 compares the data within the σ11−σ22 plane to the present model using a slope λ41122=0.041. Figure 4 presents the σ22−τ12 plane data with a parameter μ2212=−0.77(i.e., the slope of the failure surface at tau12 = 0). The model is also compared with the Gol'denblat–Kopnov4 and Cuntze12 criteria.
To test the present model with an anisotropic material with a fiber orientation distribution, we carried out experiments performed on paperboard by Suhling et al.16 Figure 5 compares our model to the strength data within the σ11−σ22 stress plane at four different shear stress levels. A very good match between model and experiments was achieved.
We validated our model against failure data from woven fabric composite materials using experiments by Gol'denblat and Kopnov.4 The present model fits the data used in the Gol'denblat and Kopnov paper very well, as shown in Figure 6. Indeed, there is sufficient data available to evaluate F1122 at three locations within the experimental data, resulting in the same interaction strength tensor component.
In summary, we have developed a new mathematically based failure criterion that is flexible enough to allow modeling of any anisotropic material. The model can easily be implemented into any finite element software to analyze results. We are currently testing the model with additive manufactured parts and plan to use it with sheet molding compound plates with various degrees of fiber orientation distribution.
- P. V. Osswald, Comparison of failure criteria of fiber reinforced polymer composites,, 2015. Technical University of Munich
- R. Talreja, Assessment of the fundamentals of failure theories for composite materials, Compos. Sci. Technol. 105, pp. 190-201, 2014.
- S. W. Tsai, Strength characteristics of composite materials,, 1965. NASA
- I. I. Gol'denblat and V. A. Kopnov, Strength of glass-reinforced plastics in the complex stress state, Mekhanika Polimerov 1, pp. 70-78, 1965.
- A. K. Malmeister, Geometry of theories of strength, Mekhanika Polimerov 2, pp. 519-534, 1966.
- S. W. Tsai and E. M. Wu, A general theory of strength for anisotropic materials, J. Compos. Mater. 5, pp. 58-80, 1971.
- P. S. Theocaris, The paraboloid failure surface for the general orthotropic material, Acta Mechan. 79, pp. 53-79, 1989.
- C. T. Sun, B. J. Quinn, J. Tao and D. W. Oplinger, Comparative evaluation of failure analysis methods for composite laminates,, 1996. DOT
- A. Puck and H. Schürmann, Failure analysis of FRP laminates by means of physically based phenomenological
models, Compos. Sci. Technol. 62, pp. 1633-1662, 2002.
- S. T. Pinho, Modelling Failure of Laminated Composites using Physically-Based Failure Models, 2005. Imperial College London
- C. G. Dávila, P. P. Camanho and C. A. Rose, Failure criteria for FRP laminates, J. Compos. Mater. 39, pp. 323-345, 2005.
- R. G. Cuntze, R. Deska, B. Szelinski, R. Jeltsch-Fricker, S. Mechbach, D. Huybrechts, J. Kopp, L. Kroll, R. Rackwitz and S. Gollwitzer, Neue Bruchkriterien und Festigkeitsnachweise für unidirektionalen Faserkunststoffverbund unter
mehrachsiger Beanspruchung---Modellbildung und Experimente Series Fortschritt-Berichte VDI: 5:506 , VDI, 1997.
- R. G. Cuntze, Aspects for achieving a reliable structural design verification?
Guiding case aerospace,, 2005. ILK Dresden
- R. G. Cuntze, Efficient 3D and 2D failure conditions for UD laminae and their application within the
verification of the laminate design, Compos. Sci. Technol. 66, pp. 1081-1096, 2006.
- P. V. Osswald and T. A. Osswald, A strength tensor based failure criterion with stress interactions, Polym. Compos., 2017.
- P. D. Soden, M. J. Hinton and A. S. Kaddour, Biaxial test results for strength and deformation of a range of E-glass and carbon fibre
reinforced composite laminates: failure exercise benchmark data, Compos. Sci. Technol. 62, pp. 1489-1514, 2002.
- J. C. Suhling, R. E. Rowlands, M. W. Johnson and D. E. Gunderson, Tensorial strength analysis of paperboard, Exper. Mechan. 25, pp. 75-84, 1985.