A Strength Tensor Based Failure Criterion with Stress Interactions

A 3D failure criterion for anisotropic materials that includes stress interactions is developed. The strength tensor model is based on the Gol’denblat–Kopnov criterion, where fourth order strength tensor stress interaction components are computed based on the slope of the failure surface at distinct stress axes intersections. Experimental data from the literature for unidirectional FRP’s, an anisotropic paperboard material and textile reinforced composites are used to validate the model.


INTRODUCTION
With the increased use of continuous, chopped and textile fiber reinforced polymer composites in the design and manufacture of critical parts and structures for the aerospace and automotive industries, interest in the strength and failure of these parts has been continually growing. In the past decades, various failure criteria models have been developed and modified. Some models were developed for unidirectional fiber reinforced composites and cannot be extended to model failure in systems neither with fiber orientation distributions nor textile reinforced composite structures. Furthermore, none of the existing models account for all modes of failure [1,2], nor include all stress interactions, if any. Some only include interactions between normal stresses, such as the interaction between longitudinal and transverse normal stresses and others only include interactions between the transverse shear stress and transverse normal stress. It can also be said that up until recently, exposure of a specific model, and even acceptance, was often linked to geographical location of the engineer.
Today, there are two general families of failure criteria. The first and oldest one are strength tensor based criteria, which represent the failure surface with a single scalar function, such as the Tsai-Hill [3], the Gol'denblat-Kopnov [4], the Malmeister [5] or Tsai-Wu [6], and the Theocaris [7] criteria. In addition to the basic engineering failure strengths such as longitudinal and transverse strengths in tension and compression, as well as shear strengths, they include the interaction between longitudinal stress and transverse stress. Such models, such as the Tsai-Wu model, are widely used and have been incorporated into finite element analysis software to perform strength analysis of anisotropic parts. However, these models do not include the interaction between normal stresses and shear stresses, which dominate the interfiber failure mode in UD-FRP's. The second family of models includes those that incorporate physical aspects of fracture, often termed phenomenological or mechanistic models, such as the ones by Sun [8], Puck [9], Pinho [10], and D avila [11]. These models do not include the interaction between longitudinal stresses and transverse stresses, but concentrate on the interaction between transverse stresses and transverse shear stresses, modeling quite well the shear strength strengthening effect during transverse compression. This second family of failure criteria was explicitly developed to predict the failure of UD-FRP.
Hence, there is still a need for a model that includes all stress interactions into one uniform equation that works for all types of anisotropic materials, such as continuous fiber, chopped fiber, and textile reinforced composite systems-a model that can be easily incorporated into finite element software in order to assess part strength. The present article derives a strength tensor based failure criteria, building on the Gol'denblat-Kopnov and Malmeister or Tsai-Wu models, which include fourth order tensor components that represent the interactions between stresses. For both failure criteria, two generalized equations are derived that describe the interactions between stresses, one for normal stress interactions, and a second for stress interactions between normal stresses and shear stresses. Finally, the new failure criterion based on the Gol'denblat-Kopnov is compared with experimental data from the literature for unidirectional fiber reinforced composite laminates, a paperboard anisotropic material and a textile-reinforced composite.

BACKGROUND
The first failure criterion for anisotropic materials appeared in Great Britain in 1948, when Hill [12] presented a model for anisotropic materials based on von Mises' distortional energy isotropic yield criteria. The 1965 Tsai adaptation of Hill's approach, now referred to as the Tsai-Hill failure criterion [3], marked the beginning of research on composites failure in the United States, and has remained one of the most widely used models to assess strength of laminated composites. While this easy to use and compact model takes into account the r 11 2r 22 stress interaction, it assumes the same strength in compression as in tension, something we know is not the case with most anisotropic materials.
The same year Tsai published the Tsai-Hill failure criterion, the Russians Gol'denblat and Kopnov [4] published a strength-tensor based failure criterion where they included tensile and compressive strengths, as well as introduced the possibility of including stress interactions. However, their model only included the r 11 2r 22 stress interaction. In 1966 in Latvia, Malmeister [5] modified the more general Gol'denblat-Kopnov model arriving at a somewhat simpler criterion, mostly thought of as the Tsai-Wu model [6]. In their 1971 article, Tsai and Wu [6] suggested various r 11 2r 22 stress interactions, but like Gol'denblat and Kopnov and Malmeister they were silent on stress interactions between shear stresses and normal stresses. Figures 1-3 compare the Malmeister or Tsai-Wu model to experimental results from the first World Wide Failure Exercise (WWFE-I) [13] in the r 11 2r 22 , r 22 2s 12 , and r 11 2s 12 planes, respectively. With the appropriate stress interaction strength tensor component, the Gol'denblat-Kopnov model predicts the same failure surfaces as the Malmeister criterion, however, the linear nature of the scalar function f in the Gol'denblat-Kopnov criterion is more conservative and seems a more reasonable approach when modeling failure. This will be discussed in more detail in the following section.
While the Gol'denblat-Kopnov and Malmeister or Tsai-Wu models inherently reflect an increase in shear strength in the transverse compressive region (Fig. 2), they do not explicitly take the shear strengthening effect, when the composite is subjected to transverse compressive loads, into account. To account for this stress interaction, Hashin [14], Sun [8], Puck [9], Cuntze [15], Pinho [10], and D avila [11], to name a few, developed various phenomenological approaches, where an internal Coulomb friction effect that increases the shear strength when a compressive transverse stress is applied, is included. Hashin was the first to include such an effect, by treating fiber failure and inter-fiber failure differently. With this, he laid down the groundwork for various failure criteria that followed its lead in the subsequent three decades. Sun [8]  factor was included by controlling the slope of the failure surface in the r 22 2s 12 plane at r 22 50 in the compressive transverse region, basically introducing a well defined r 22 2s 12 interaction. Both Hashin and Sun used a single matrix strength Y, not distinguishing between a tensile Y t and a compressive Y c inter-fiber failure strength, reducing the degrees of freedom when trying to fit experimental data. Furthermore, none of the models reflected an observed r 11 2s 12 stress interaction that leads to a shear strengthening effect observed during longitudinal tension, as presented in Fig. 3 [13].
Other well-known phenomenological models with r 22 2s 12 stress interactions have been developed over time.
In 1996, the German composites researcher Puck [9] proposed the first true phenomenological laminate failure approach, built on his own experience in the failure of composite materials [16] and on Hashin and Sun's approaches. While Puck's "action plane" model does a very good job when including the shear strengthening effect under transverse stresses, as shown in Fig. 2, and predicting the angle of the fracture surface in the composite, the model depends on several fitting parameters that were developed specifically for the failure of unidirectional fiber reinforced laminate materials, and cannot be extended to model failure of other composites such as chopped fiber or textile reinforced composites. Puck also introduces several factors that require significant experience for its implementation and therefore make the model more difficult to use. Puck's model does not include the r 11 2r 22 stress interaction and therefore, as shown in Fig.  1, does a poor job when trying to match the experimental failure surface data in the r 11 2r 22 plane. Other notable failure criteria that use a phenomenological approach are the models by Pinho [10] in Great Britain, D avila et al. [11] in the United States and Cuntze [15] in Germany. They all help in the understanding of inter-fiber failure phenomenological behavior and predict experimental measurements in the r 22 2s 12 plane quite well. Cuntze et al. [17] performed an extensive study on the failure of UD-FRP composites, demonstrating various effects resulting from stress interactions. Ultimately, his work led to his failure criterion that incorporates the Coulomb internal friction effect for the inter-fiber failure modes, but also couples the fiber failure and inter-fiber failure criteria [15]. Similar to the Gol'denblat-Kopnov and Malmeister or Tsai-Wu models, Cuntze's model, with a certain manipulation of tensile and compressive stresses, can be expressed with a single scalar function f , therefore coupling the failure surface to the three-dimensional stress field. Cuntze's model did not include other stress interactions, and therefore does not predict well the failure surface in the r 11 2r 22 plane. The stress invariants used in Cuntze's criterion were derived for transversely isotropic composites, such as UD-FRP's, however, he has also published the stress invariants to be used when modeling transverse anisotropic materials, such as chopped fiber composites and textile reinforced composites [18].

STRENGTH TENSOR BASED FAILURE CRITERIA
The failure criterion proposed in this article is based on strength tensor based models such as the Gol'denblat-Kopnov and the Malmeister or Tsai-Wu models. The model presented by Gol'denblat and Kopnov defines a scalar failure function as a function of strength tensors and stresses using were failure is expected when f 1. The proposed strength tensor components F ij , F ijkl , and F ijklmn are second, fourth and sixth order tensors, respectively, that depend on engineering strength parameters such as X t , X c , Y t , Y c , and S; and satisfy the symmetry conditions, F ij 5 F ji and F ijkl 5F klij . However, in their analysis they only included the second and fourth order tensor terms, since higher order stress tensors, such as sixth or eighth, would increase the number of tensor components into the hundreds and thousands, respectively. To achieve a linear criterion scalar function f , Gol'denblat and Kopnov set the exponents a51, b51=2, and g51=3.
where the usual notation of r ij and s ij , for normal and shear stresses, respectively, was used. Gol'denblat and Kopnov used Eq. 2, without the stress interaction terms F 1112 and F 2212 , to solve for most of the strength tensor components by assuming uniaxial conditions in the 1 and 2 directions or a shear condition in the 1-2 plane. These are all listed in Table 1. In order to determine the r 11 2r 22 interaction strength tensor component F 1122 , Gol'denblat and Kopnov measured the positive and negative shear strengths, S 45p and S 45n , of a specimen where the fibers where oriented at 45 . The resulting F 1122 stress interaction term is also listed in Table 1.
Malmeister [5] and Tsai and Wu [6] modified the Gol'denblat-Kopnov model by letting a51 and b51 from the equation, eliminating the fractional exponents, but resulting in a function f that is quadratic with respect to stress. Figure 4 illustrates this effect by comparing both models to WWFE-I data for the r 22 2s 12 failure plane. One can see that both models have the same failure surface, f 51, but by delivering a failure function f that is linear with respect to the applied stress field, the Gol'denblat-Kopnov is a more conservative approach. Furthermore, a linear function can be more easily implemented when using probabilistic failure analysis as for example proposed in the Soviet Union in 1975 by Zaitsev et al. [19] and Thieme et al. [20] in Germany in 2014. Even when using a simple factor of safety approach, in the example depicted in Fig. 4 the Gol'denblat-Kopnov criterion has no failures when using a safety factor above 1.15 (f 50:87), whereas the Tsai-Wu approach would still predict a failure at a safety factor of 2.1 (f 50:47). Table 1 presents the strength tensor components for both the Gol'denblat-Kopnov and the Malmeister or Tsai-Wu models. The r 11 2r 22 interaction strength tensor component F 1122 proposed by Tsai and Wu [6], listed in Table 1, depends on the factor f 12 , which lies between 21 and 0. A popular approach is letting f 12 521=2, because it leads to the classic von Mises theory for isotropic materials [21]. However, these models neglect the longitudinal and transverse interaction terms, F 1112 and F 2212 , which represent the r 11 2s 12 and r 22 2s 12 interactions. These interactions, as well as a new r 11 2r 22 interaction are proposed in this article and are presented in the next section.

STRESS INTERACTION STRENGTH TENSOR COMPONENTS
In this article we propose interaction strength tensor components, F ijkl , based on the slopes of the failure surface (f 51) at any of the points where the engineering strength values are known within an arbitrary r ij 2r kl plane. With minimal experimental data gathered to compute the slope of the failure surface around any of the four strength values within a r ij 2r kl plane, the interaction strength tensor component F ijkl can be evaluated. For this, we derive the interaction terms between normal stresses and the interaction between normal and shear stresses separately, as schematically depicted in Fig. 5.
To illustrate this we can take the r 11 2r 22 interaction, F 1122 , point on the failure surface where r 11 5r u 112t 5X t and r 22 50 and the failure surface has a slope of F 1122 can be evaluated at any of the remaining three axes intersections of the failure surface within the r 11 2r 22 plane, theoretically resulting in the same computed numerical value for the strength tensor interaction term. Table 2 presents the interaction strength tensor components for the more general case where the subscripts 1122 where replaced with iijj.
To represent the interaction between normal stresses and shear stresses, such as F 2212 , we assume symmetry (replacing s ij by js ij j) and propose an interaction term F iiij The r 22 2s 12 strength tensor interaction term can be written in terms of engineering strength values as The term is also presented in general form in Table 2 for both failure criteria.

COMPARISON WITH EXPERIMENTAL DATA
The present model, based on the Gol'denblat-Kopnov criterion was compared with other models and experimental data for unidirectional FRP materials, as well as experimental data for an anisotropic paperboard and a two-ply plain weave CFRP laminate.

Unidirectional FRP
The failure of unidirectional composites in the r 11 2r 22 , s 12 2r 22 , and s 12 2r 11 stress planes was analyzed with the present model, and compared with experiments performed with glass and carbon reinforced epoxy composites. The data was taken from the first World Wide Failure Exercise (WWFE-I) [13].
The WWFE-I fiber failure data for the r 11 2r 22 plane presented in Fig. 1 was modeled using the present model using a k 1112 4 5 0.041 and presented in Fig. 6. The resulting failure surface, f 51, is identical to the failure surface predicted by the Malmeister or Tsai-Wu criteria using the interaction coefficient f 12 521=2.
The second test of the model was done with the interfiber failure WWFE-I data s 12 2r 22 stress plane used in Figs. 2 and 4. The best fit of the experimental data resulted with a parameter l 2212 5 20.77. The present model is compared with the Gol'denblat-Kopnov and the Cuntze criteria in Fig. 7. For the Cuntze criterion l 2212 2t 5l 2212 2c 5 20.55 was used. The Cuntze "friction" parameter can be adjusted independently in tension and compression to fit the data accordingly. However, the Cuntze model misses the data points on the left of the curve. Here, the experimental data exhibits a compressive transverse strength increase that is also reflected by the present model. It is not quite clear if the compressive transverse strength increase is real. While the authors Fii1Fjj k iijj 4 22Fjjjj r u jj2c k iijj have observed this "bulge" in other experimental data [17], the literature remains silent. Figure 8, presents another set of measurements that resulted from an extensive experimental study performed by Cuntze et al. [17], The last set of experimental data used to test the present model with unidirectional FRPC materials, is the longitudinal stress under shear stress in the s 12 2r 11 stress plane also used in Fig. 3. Using the present model with a parameter l 1112 5 0.054, the shear strengthening effect under tensile longitudinal loads is clearly observed in Fig. 9.

Anisotropic Paperboard
To test the present model with an anisotropic material with a fiber orientation distribution, experiments performed on paperboard by Suhling et al. [22] were used. In their study, they found that when s 12

Two-Ply Plain Weave CFRP Laminate
To validate the present model against failure data from woven fabric composite materials, experiments by Mallikarachchi and Pellegrino [23] were used. The Gol'denblat and Kopnov data presented sufficient information to show that the r 11 2r 22 plane failure surface does not have a tilt and is therefore relatively circular. This is in agreement with the failure criterion developed in Greece by Theocaris [7] as well as his experimental work dealing with weaves [24]. The present model can fit the data used in the Gol'denblat and Kopnov article very well, as can be seen in Fig. 11. There is sufficient data available to evaluate F 1122 at three locations in the Gol'denblat and Kopnov experimental data for a textile composite [4]. The authors computed the three slopes which resulted in To validate the present model against failure data from woven fabric composite materials with a combined longitudinal, transverse and planar shear stresses, experiments performed by Mallikarachchi and Pellegrino [23] on twoply T300-1k/Hexcel 913 plain weave laminates were used. In their own analysis, Mallikarachchi and Pellegrino used the Tsai-Wu model with a four-fold symmetry about the third axis of the laminate, resulting in F 11 5F 22 , F 1112 5F 2212 , etc. and used Tsai-Wu's F 1112 longitudinaltransverse stress interaction tensor component with f 12 52 1 2 . As a result, as shown in Fig. 12, their failure surface was elliptical, contradicting the findings of Gol'denblat and Kopnov [4] as well as Theocaris [24]. The present model was fit in such a way that it predicts a nearly circular failure surface by using k 1112 4 5 20.35. Furthermore, the parameters l 1112 and l 2212 where adjusted such that the two average data points within the r 11 2r 22 failure plane, located at shear stress s 12 592:6, fell on the failure surface. The values of l 1112 5 l 2212 5 0.102 put the failure surface on top of the average values resulting in a perfect fit of the available data.

CONCLUSIONS
A strength tensor based failure criterion with stress interactions based on the Gol'denblat-Kopnov criterion was developed and tested. The proposed failure criterion with its stress interaction strength tensor components accurately predicts failure envelopes and failure trends for unidirectional fiber reinforced composite materials, as well as for anisotropic materials with a certain fiber orientation and woven fabric fiber reinforced composites. The criterion was able to capture fiber failure and interfiber failure phenomena for unidirectional FRP's. Furthermore, the model is able to introduce the shear strengthening effect when the unidirectional composite is subjected to longitudinal tensile loads. The model's failure surface was able to accurately predict measurements performed on an anisotropic paperboard. Finally, the proposed criterion was used to accurately predict failure within a woven carbon fiber fabric reinforced laminate. While the proposed model ignores phenomenological, micromechanical aspects of composites failure, it presents an alternative mathematically based technique, that can be easily implemented and used in conjunction with strength assessment of complex anisotropic parts.