# An improved strength tensor-based failure criterion with stress interactions

February 1, 2017

**Author(s):** Meryem Kalkan Erdogan, Meral Karakisla, and Mehmet Saçak

**Year:** 2017

*A new failure criterion includes tensor stress interaction components that can be computed using minimal experimental data depicting the slope of the failure surface at distinct stress axes intersections.*

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, 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. Others only include interactions between transverse shear stress and transverse normal stress.^{}

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.

Figure 1.

Types of fiber-reinforced polymer composites. τ_{ij}: Shear stresses acting in the i direction on a face with a normal j direction. σ_{ii}: Normal stresses in the i direction on a face with a normal i direction. i and j can be 1, 2, and 3, representing local material x, y, and z coordinates.

The failure criterion presented here is a strength tensor model based on the Gol'denblat-Kopnov criterion. 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 as

_{12}=|τ

_{12}|. Here, F

_{11}, F

_{22}, and F

_{12}are second-order strength tensors and F

_{1111}, F

_{2222}, F

_{1212}, and so forth are fourth-order strength tensors. σ

_{11}and σ

_{22}are normal stresses, and τ

_{12}represents shear stress. The strength tensor components required in the above equation, and originally derived by Gol'denblat and Kopnov, are given by

_{t}, X

_{c}, Y

_{t}, and Y

_{c}are tensile (t) and compressive (c) strengths in the longitudinal (X) and transverse (Y) directions, and S represents the planar shear strength. Our model includes all interaction terms as long as sufficient data exists, which in the above equation includes F

_{1122}, F

_{1112}, and F

_{2212}. These interaction terms are based on the slopes of the failure surface (f=1) at any of the points where the engineering strength values are known within an arbitrary plane, as schematically depicted in Figure 2. 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 σ

_{11}−σ

_{22}plane. To compute F

_{1122}, one may choose any of the following equations:

_{22}−τ

_{12}plane, one slope can be used and the interaction term is given by

Locations on the failure surface within the (a) σ_{11}- σ_{22} and (b) σ_{22}- τ_{12} planes where the interaction F_{122} and F_{2212}, respectively, are evaluated. λ_{1 - 4}: Slopes at the four axes intersections. μ_{2212}: Slope of the failure surface at τ_{12}=*0*. X_{t}, X_{c}, Y_{t}, and Y_{c}: Tensile (t) and compressive (c) strengths in the longitudinal (X) and transverse (Y) directions. S: Planar shear strength.

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 σ_{11}−σ_{22} and σ_{22}−τ_{12} stress planes using our model, and compared it with first World Wide Failure Exercise (WWFE-I) experiments performed with glass-fiber-reinforced epoxy composites. Figure 3 compares the data within the σ_{11}−σ_{22} plane to the present model using a slope λ_{4}^{1122}=0.041. Figure 4 presents the σ_{22}−τ_{12} plane data with a parameter μ^{2212}=−0.77(i.e., the slope of the failure surface at tau_{1}2 = 0). The model is also compared with the Gol'denblat–Kopnov and Cuntze criteria.

Figure 3.

Comparing the present model using λ_{4}^{1122}=*0.041*with the first World Wide Failure Exercise (WWFE-I) experimental data.

Figure 4.

Comparing the present model using μ^{2212}=*- 0.77* with a biaxial failure stress envelope under transverse stress and shear stress loading data from WWFE-I and with the Cuntze model.

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. Figure 5 compares our model to the strength data within the σ_{11}−σ_{22} stress plane at four different shear stress levels. A very good match between model and experiments was achieved.

Figure 5.

Comparing the present model with biaxial in-plane strength results for paperboard experimental results using λ_{4}^{1122}=*- 0.21*, μ^{1112}=*0.25*, and μ^{2212}=*0.20*. (a) Failure surface when τ_{12}=*0*. (b) Failure surface when τ_{12}=*6.9MPa*. (c) Failure surface when τ_{12}=*10.3MPa*. (d) Failure surface when τ_{12}=*15.9MPa*.

We validated our model against failure data from woven fabric composite materials using experiments by Gol'denblat and Kopnov. 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 F_{1122} at three locations within the experimental data, resulting in the same interaction strength tensor component.

Figure 6.

Comparing the present model with biaxial in-plane strength results for a weave glass-fiber-reinforced polymer (GFRP) laminate, with either λ_{1}^{1122}=*- 2.82*, λ_{2}^{1122}=*- 0.38*, or λ_{4}^{1122}=*0.38*, which all give F_{1122}=*1.8*×*10*^{- 8}(MPa)^{-1}.

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.

## Authors

**Tim A. Osswald**

University of Wisconsin-Madison

Tim A. Osswald is a professor of mechanical engineering at the University of Wisconsin-Madison, where he co-directs the Polymer Engineering Center. He is also an honorary professor at Friedrich Alexander University in Erlangen, Germany, and at the National University of Colombia, Bogotá.**Paul V. Osswald**

Technical University of Munich

Paul V. Osswald received his BSc in composite materials engineering at Winona State University, Minnesota, and is currently pursuing his MSc in mechanical engineering at the Technical University of Munich.

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**DOI:** 10.2417/spepro.006867

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