Enhancing Prediction Accuracy In Sift Theory J. Wang 1 *, W. K. Chiu - PDF document

18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS Enhancing Prediction Accuracy In Sift Theory J. Wang 1 *, W. K. Chiu 2 1 Defence Science and Technology Organisation, Fishermans Bend, Australia, 2 Department of Mechanical & Aerospace Engineering, Monash University, Clayton, Australia * Corresponding author ( john.wang@dsto.defence.gov.au ) Keywords : SIFT, matrix failure, polymer composite, failure prediction 1 Introduction These two strain variants can indicate matrix initial failure due to volume increase (dilational strain) and In the conventional laminate theory widely used to distortional strains respectively. When either of these predict the strength of fibre-reinforced polymer reaches its critical value, failure will occur. Note that composites, the laminae are treated as homogeneous Equation 2 is essentially Von Mises equivalent strain. orthotropic materials. Gosse and Christensen [1] This paper aims to demonstrate the accuracy of matrix adopted a micromechanical approach which predicts failure prediction could be increased significantly by separately the failure of the polymer matrix and enhancing these two failure criteria. fibres 1 . They used strain invariants as failure criteria 2 Discussion about Polymer Failure Criteria and thus this approach was named as Strain Invariant Failure Theory (SIFT). In the following discussion four basic load cases will In the application of SIFT, a microstructure analysis is be considered, namely uni-axial compression, pure in- conducted on a unit cell of the composite material that plane shear, uniaxial tension and biaxial tension, refer contains a fibre and surrounding polymeric matrix to to Figure 1. In most part of this paper, only the initial determine the relationship between the stress-strain matrix failure is considered, that is the discussion is states of the whole cell and its matrix and fibre restricted to the linear elastic condition. (The components. This relationship is then used in a application to prediction of matrix ultimate failure structural analysis to predict matrix or fibre failure. In will be briefly discussed in Section 2.4.) The stress a linear finite element method, the SIFT analysis and strain states under the four load cases are: • Uniaxial compression could be implemented during post processing the σ 1 <0, σ 2 = σ 3 =0; ε 1 = σ 1 /E <0, ε 2 = ε 3 = - λε 1 results from a computation based on the conventional • Pure in-plane shear laminate theory, by correlating the element stress- σ 1 >0, σ 2 =- σ 1, σ 3 =0; ε 1 =(1+ λ ) σ 1 /E > 0, ε 2 =- ε 1 , ε 3 = 0 strain state with matrix and fibre stress-strain states. • Uniaxial tension This theory has been used by a number of researchers σ 1 >0, σ 2 = σ 3 =0; ε 1 = σ 1 /E >0, ε 2 = ε 3 = - λε 1 and some success has been reported [2]. For the matrix failure prediction, Gosse and • Biaxial tension σ 1 = σ 2 >0, σ 3 =0; ε 1 = ε 2 =(1- λ ) σ 1 /E , ε 3 = -2 λε 1 Christensen proposed that two properties that control damage in the matrix, are the first invariant of the where σ 1 , σ 2 and σ 3 are principal stresses, E is the J , Young’s modulus and λ is the Poison’s ratio. strain tensor, and the second invariant deviator, 1 ε ε : σ 1 σ 1 σ 1 σ 1 eqv = ε + ε + ε J σ 2 σ 2 (1) ε 1 1 2 3 { } ε = ε − ε + ε − ε + ε − ε 0 . 5 2 2 2 0 . 5 [( ) ( ) ( ) ] (2) eqv 1 2 1 3 2 3 ε ε ε , , where are principal strains. 1 2 3 Fig.1. Four basic load cases considered 2.1 Uni-axial Compression and Pure Shear Cases In terms of the SIFT theory, the failure criterion based 1 In principle this micromechanical approach may also on Equation 2 applies to these two cases. predict the interfacial failure between the fibre and matrix.

It is well known that the strength of polymer materials Note the following relationship: + λ could be affected by compressive hydrostatic stress. 1 ε = σ (7) For those polymer materials, to cover both uniaxial eqv eqv E compression and pure shear load cases, a yield − λ 1 2 criterion such as the Drucker-Prager criterion 2 [3] = J J (8) ε σ 1 1 E would be more suitable than Von Mises criterion. The failure prediction equations can alternatively be Table 1 provides yield strength data for two typical expressed using stress variables with full equivalence. resin materials used in laminate composites. For isotropic materials, a stress based failure criterion is more commonly used since it is often easier to use Table 1. Typical resin material strength data [4] and have clearer physical meaning (e.g. hydrostatic Resin Tensile Compression Shear (MPa) (MPa) (MPa) pressure effect shown in Drucker-Prager criterion). Type 1 Yield 58 96 50 Alternative equations of Equations (3) and (4) strength expressed in stresses are: E =3.6GPa σ + = Ultimate 58 130 62 BJ A (9) σ λ =0.35 eqv 1 strength σ = σ + = BJ A ' (10) Type 2 Yield 50 100 36 σ eqv eqv 1 Strength For the materials listed in Table 1, A and B values E =3.9GPa Ultimate 50 130 60 calculated are listed in Table 2. λ =0.35 strength Table 2. Parameter A and B values For the prediction of yield strength, if we generate the Resin Strain based equation Stress based equation ε critical value of from pure shear data, the A B A B eqv 32.5x10 3 µε measured yield strength in the uni-axial compression Type 1 0.439 86.6MPa 0.098 21.6x10 3 µε case will be higher than the predicted using Equation Type 2 1.69 62.4MPa 0.376 2 by 11% and 60% respectively for these two materials. This discrepancy can be removed by 2.2 Uni-axial and Biaxial Tension Cases applying the 2-parameter Drucker-Prager criterion Before discussing the uni-axial and biaxial tension (based on Equation 3). ε + = BJ A cases, we may consider an extreme case where the (3) eqv ε 1 polymer is loaded with uniformly distributed tensile ε J are calculated using Equations 1 where and eqv stress in all the three axial directions. It is well known 1 that the Von Mises yield criterion is not valid in this and 2 with the measured failure data from compression and shear tests (Table 1); A and B are situation. A Drucker-Prager criterion established using parameters determined from compression and shear two parameters determined from this equation. load cases would also significantly over-predict the Failure is predicted when: ε = ε + = BJ A strength in this situation. One commonly adopted ' (4) ε eqv eqv 1 approach is to use a maximum tensile stress or strain ε ' where is the revised equivalent strain. Equations criterion to handle such tensile dominated load eqv conditions, in conjunction with other criteria such as (5) and (6) below describe the first invariant of the J Von Mises or Drucker-Prager to handle other load stress tensor, , and the second stress invariant 1 σ conditions. In contrast, Equation 1 adopted by Gosse σ deviator, : eqv and Christensen contains all the 3 principal strain = σ + σ + σ J components in linear combination that indicates the (5) σ 1 1 2 3 { volume increase of the material. } σ = σ − σ + σ − σ + σ − σ 0 . 5 2 2 2 0 . 5 [( ) ( ) ( ) ] (6) Bardenheier [5-6] provided experimental data from eqv 1 2 1 3 2 3 uni-axial and biaxial tension tests of three types of polymer materials. Analysing these data, it is 2 Only basic failure criteria that are relatively easy to indicated that under a biaxial tensile load condition, implement are considered in this study. prediction based on Equation 1 would significantly 2