A NEW STRENGTH TENSOR BASED FAILURE CRITERION FOR ANISOTROPIC MATERIALS

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1 A NEW STRENGTH TENSOR BASED FAILURE CRITERION FOR ANISOTROPIC MATERIALS Paul V. Osswald, Technical University of Munich, Munich, Germany Tim A. Osswald, Polymer Engineering Center, University of Wisconsin-Madison, WI Abstract A recently published 3D failure criterion [] for anisotropic materials that incorporates stress interactions is presented. The strength tensor model includes fourth order strength tensor stress interaction components that can be computed based on the slope of the failure surface at distinct stress axes intersections, making the model apt to fit failure surfaces for a variety of anisotropic materials. Experimental data from the literature for unidirectional FRP s, an anisotropic paperboard material and textile reinforced composites are used to validate the model. 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 [9], Puck [0], Pinho [], Dávila [2] and Cuntze [3-5]. 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. Introduction With rapidly growing 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 with fiber orientation distributions, nor textile reinforced composite structures. Furthermore, none of the existing models account for all modes of failure [2,3], 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. 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 [4], the Gol denblat- Kopnov [5], the Malmeister [] or Tsai-Wu [7] and the Theocaris [8] 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 don t include the interaction between normal stresses and shear stresses, which dominate the inter-fiber Figure : Types of FRP Composites This paper presents a strength tensor based failure criterion, recently published by the authors [], which includes fourth order tensor components that represent the interactions between stresses and can be applied to any anisotropic fiber reinforced composite material, such as continuous, chopped and woven fiber reinforced systems, subjected to complex stress fields, as schematically depicted in Figure. The new failure criterion is SPE ANTEC Anaheim 207 / 74

2 compared to experimental data from the literature for unidirectional fiber reinforced composite laminates, a paperboard anisotropic material and a textile-reinforced composite. Strength Tensor Based Failure Criterion The failure criterion presented here [] is a strength tensor model based on the Gol denblat-kopnov criterion [5]. The model presented by Gol denblat and Kopnov defines a scalar failure function, f, as a function of strength tensors and stresses using f = F %& σ %& ( + F%&*+ σ %& σ *+, + () were failure is expected when f. The proposed strength tensor components F %& and F %&*+ are second and fourth order tensors, respectively, that depend on engineering strength parameters such as X, X 2, Y, Y 2, and S, which represent the maximum tensile and compressive stresses for the longitudinal (σ 55 ) and transverse (σ ) directions and maximum shear strength (τ 5 ), respectively. The tensors satisfy the symmetry conditions, F %& = F &% and F %&*+ = F *+%&. To achieve a linear criterion scalar function f, Gol denblat and Kopnov set the exponents α = and β = /2. Through this additive tensor technique Gol denblat and Kopnov were able to couple all the failure modes into one single function, where a separate treatment of compressive and tensile modes is not required. For the plane stress case the Gol denblat-kopnov criterion becomes f = F 55 σ 55 + F σ + F 5 τ 5 + F 5555 σ 55 + F σ + F 55 τ 5 + 2F 55 σ 55 σ + 2F 555 σ 55 τ 5 + 2F 5 σ τ 5 5 (2) where the usual notation of σ %% and τ %&, for normal and shear stresses, respectively, was used. Gol denblat and Kopnov dropped the stress interaction terms F 555 and F 5 terms and solved for most of the strength tensor components by assuming uniaxial conditions in the and 2 directions or a shear condition in the 2 plane. In order to determine the σ 55 σ interaction strength tensor component F 55, Gol denblat and Kopnov measured the positive and negative shear strengths, S =>? and S =>@, of a specimen where the fibers where oriented at 45. The strength tensor terms given by Gol denblat and Kopnov [5] are presented in Table. The model developed by the authors includes interaction terms F 555 and F 5 as well as the general strength tensor components, F %&*+. These interaction terms are based on the slopes of the failure surface (f = ) at any of the points where the engineering strength values are known within an arbitrary σ %& σ *+ plane. Hence, there is minimal experimental data needed to compute the slope of the failure surface around any of the four strength values within a σ %& σ *+ plane, hence, being able to calculate the interaction strength tensor component F %&*+. For this, the interaction terms between normal stresses and the interaction between normal and shear stresses are derived separately, as schematically depicted in Figure 2. Table. Strength tensor components Tensor From Gol denblat-kopnov [5] Component F 55 2 X X 2 F F 2 F 4 X + X 2 Y Y 2 Y + Y 2 F 5 0 F 55 S F 55 8 X + X 2 + Y + Y2 S =>? + S =>@ To illustrate this we can take the σ 55 σ interaction, F 55, point on the failure surface where F σ 55 = σ 55E = X and σ = 0 and where the failure surface has a slope of HI JJ = λ 55 HI 5. After taking the KK derivative of equation (2) with respect to σ 55, at the failure surface where f = and τ 5 = 0, the interaction term can easily be computed and is given by K F 55 = N KKKKJ N KK ON JJ P KKJJ K ONKKKK PKKJJ (5) K F 55 can also be evaluated at any of the remaining three axes intersections of the failure surface within the σ 55 σ plane. Ideally, the same computed numerical value for the strength tensor interaction term will result from all four axes intersections. Table 2 presents the interaction strength tensor components for the more general case where the subscripts 22 where replaced with iijj. To mathematically evaluate the interaction between normal stresses and shear stresses, such as F 5, SPE ANTEC Anaheim 207 / 742

3 we assume symmetry by replacing τ %& with τ %& and propose an interaction term F %%%& based on the slope of the failure surface, HS TU HI TT = μ %%%&, at τ %& = τ %& F = S and σ %% = 0, as schematically depicted in Figure 2. This strength tensor component is also presented in general form in Table 2. The reader should consult reference [] for the Malmeister or Tsai-Wu models interaction strength tensor components derived using the above technique. anisotropic paperboard and a two-ply plain weave CFRP laminate. Table 2. Interaction strength tensor components (τ F 5 = S, etc.). Tensor Present model Component F %%%& F %% τ %& F F %&%&μ %%%& F (λ 5 ) F (λ ) F %% + F && λ 5 F %%%% 5/ + F %%%% λ 5 F %% + F && λ F &&&& 5/ F &&&& λ F (λ Y ) F %% + F && λ Y F %%%% 5/ F %%%% λ Y F (λ = ) F %% + F && λ = F &&&& 5/ F &&&& λ = Unidirectional FRP The failure of unidirectional composites in the σ 55 σ, τ 5 σ and τ 5 σ 55 stress planes was analyzed with the present model, and compared to experiments performed with glass and carbon reinforced epoxy composites. The data was taken from the first World Wide Failure Exercise (WWFE-I) []. The WWFE-I fiber failure data for the σ 55 σ plane presented in Figure 3 was modeled using the present model using a λ = 555 =0.04. Figure 2: Locations on the failure surface within the σ %% σ && (a) and σ %% τ %& (b) planes where the interaction F and F %%%&, respectively, are evaluated Comparison with Experimental Data The present model was tested by comparing it to other models and experimental data for unidirectional FRP materials, as well as experimental data for an Figure 3: Comparing the present model using λ = 555 =0.04 to WWFE-I experimental data The second test of the model was done with the inter-fiber failure WWFE-I data τ 5 σ stress plane used in Figures 4. The best fit of the experimental data resulted with a parameter μ 5 = The present model is also compared to the Gol denblat-kopnov [5] and the Cuntze SPE ANTEC Anaheim 207 / 743

4 [3] criteria. For the Cuntze criterion μ 5E = μ 5E2 = 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 have observed this bulge in other experimental data [3], the literature remains silent on this issue. Figure 4: Comparing the present model using μ 55 =-0.77 to a biaxial failure stress envelope under transverse stress and shear stress loading data from the WWFE-I [] and to the Cuntze model 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 τ 5 σ 55 stress plane also used in Figure 5. Using the present model with a parameter μ 555 =0.054, the shear strengthening effect under tensile longitudinal loads is clearly observed in the figure. Figure 5: Comparing the present model to WWFEI biaxial failure stress envelope data under longitudinal and shear stress loading results [], using μ 555 =0.054 Anisotropic Paperboard To test the present model with an anisotropic material with a fiber orientation distribution, experiments performed on paperboard by Suhling et al. [7] were used. Figure : Comparing the present model to biaxial inplane strength results for paperboard experimental results [7] using λ = 55 =-0.2, μ 555 =0.25 and μ 5 =0.20 SPE ANTEC Anaheim 207 / 744

5 In their study, they found that when τ 5 =0, the Tsai-Wu criterion with F 55 = E= (MPa) - gave the best fit. However, when including all four levels of shear, τ 5 =0, τ 5 =.9 MPa, τ 5 =0.3 MPa and τ 5 =5.9 MPa, they had to drop the longitudinal - transverse stress interaction tensor component (F 55 = 0) to achieve an overall better fit. This compromised somewhat the results at τ 5 =0, because the tilt of the elliptical failure surface and observed in the experimental results is lost when no stress interaction exists. On the other hand, in addition to including the stress interaction strength tensor F 55, the present model is able to include both shear stress - normal stress interactions F 555 and F 5, by adjusting λ 555 =, μ 555 and μ 5, respectively. Figure compares the strength data within the σ 55 σ stress plane at the four different shear levels to the present model. It is clear that by including all three stress interaction strength tensor components, a very good match between model and experiments was achieved. Figure : Comparing the present model to biaxial in-plane strength results for paperboard experimental results [7] using λ = 555 =-0.2, μ 555 =0.25 and μ 5 =0.20 and µ 5 where adjusted such that the two average data points within the σ 55 σ failure plane, located at shear stress τ 5 = 92., fell on the failure surface. The values of μ 555 = μ 5 =0.02 put the failure surface on top of the average values resulting in a perfect fit of the available data. Two-ply plain weave CFRP laminate To validate the present model against failure data from woven fabric composite materials, experiments by Mallikarachchi and Pellegrino [8] were used. The Gol denblat and Kopnov data presented sufficient information to show that the σ 55 σ 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 [8] as well as his experimental work dealing with weaves [9]. The present model can fit the data used in the Gol denblat and Kopnov paper very well, as can be seen in Figure 7. There is sufficient data available to evaluate F 55 at three locations in the Gol denblat and Kopnov experimental data for a textile composite [5]. The authors computed the three slopes which resulted in the same longitudinal transverse stress interaction tensor component of approximately F 55 =.8 0 E` (MPa) -. 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 [8] on two-ply T300-k/ Hexcel 93 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 55 = F, F 555 = F 5, etc. and used Tsai-Wu s F 555 longitudinal - transverse stress interaction tensor component with f 5 = 5. As a result, as shown in Figure 8, their failure surface was elliptical, contradicting the findings of Gol denblat and Kopnov [5] as well as Theocaris [9]. The present model was fit in such a way that it predicts a nearly circular failure surface by using λ 555 = = Furthermore, the parameters µ 555 Figure 7: Comparing the present model to biaxial in-plane strength results for a weave GFRP laminate for τ 5 = 0 [5], with either λ =-2.82, λ 555 = or λ = 555 =0.38 which all give F 55 =.8 0 E` (MPa) - Conclusions The new strength tensor based failure criterion with stress interactions based on the Gol denblat-kopnov criterion was successfully compared with 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 inter-fiber 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. SPE ANTEC Anaheim 207 / 745

6 Figure 8: Comparing the present model to biaxial in-plane strength results for a two-ply plain weave CFRP laminate for τ 5 = 0 (right) and τ 5 = 92. MPa (right) [8], with λ = 555 =-0.35 and μ 555 = μ 5 = S.W. Tsai. Strength characteristics of composite materials. NASA CR-224, (95) I.I. Gol denblat, V.A. Kopnov, Mekhanika Polimerov,, (95)... A.K. Malmeister, Mekhanika Polimerov, 4, , (9) S.W. Tsai, E.M. Wu, J Comp Mater, 5, (97) P.S. Theocaris, Acta Mechanica, 79, (989) C.T. Sun, B.J. Quinn, D.W. Oplinger, Comparative Evaluation of Failure Analysis Methods for Composite Laminates. DOT/FAA/AR-95/09, (99) A. Puck, H. Schürmann, Compos Sci Technol, 58, (998)... S.T. Pinho, Modeling failure of laminated composites using physically based failure models. Ph.D. Thesis, Department of Aeronautics, Imperial College London, (2005) C.G. Dávila, P.P. Camanho, C.A. Rose, J Comp Mater, 39, (2005) R.G. Cuntze, R. Deska, B. Szelinski, R. Jeltsch-Fricker, S. Mechbach, D. Huybrechts, J. Kopp, L. Kroll, S. Gollwitzer, R. Rackwitz. Neue Bruchkriterien und Festigkeitsnachweise für unidirektionalen Faserkunststoffverbund unter mehrachsiger Beanspruchung Modellbildung und Experimente. VDI Fortschrittbericht. 5.50, R.G. Cuntze. Aspects for achieving a reliable structural design verification? Guiding case aerospace. Großer Seminarvortrag ILK Dresden. (2005) R.G. Cuntze, Compos Sci Technol,, (200)... P.D. Soden, M.J. Hinton, A.S. Kaddour, Compos Sci Technol, 2, (2002) J.C. Suhling, R.E. Rowlands, M.W. Johnson, D.E. Gunderson, Experim Mech, 25, (98) H.M.Y.C. Mallikarachchi, S. Pellegrino, Journal of Composite Materials, 47, (203) P.S. Theocaris, Acta Mechanica, 95, 8-8 (992). References.. P.V. Osswald and T.A. Osswald, Polymer Composites, DOI:0.002/pc.24275, (207) R. Talreja, Compos Sci Technol, 05, (204) P.V. Osswald. Comparison of Failure Criteria of Fiber Reinforced Polymer Composites. Term Thesis, Technical University of Munich, TUM MW5/520 SA, (205). SPE ANTEC Anaheim 207 / 74