u ¼ 4F π s11 s2 13 s33 0:19þsinh 1 r= b ð2:1Þ b2 ¼ 4Fr π 1 E1 ν2 31 E3 ð 2:2Þ where, F—load per unit length along the fiber longitudinal direction u—platen displacement L—length of the compressed fiber r—radius of the fiber b—contact half width E1—Young’s modulus in 1–2 plane of transverse isotropy E3—longitudinal Young’s modulus in the fiber direction w—compressed half width s11 ¼ 1 E1 s12 ¼ ν12 E1 s13 ¼ ν13 E1 s33 ¼ 1 E3 The intrinsic material behavior is isolated from the total response by defining true stress as σ ¼ F 2weff where the effective widthweff ¼ wþb 2 . The original fiber cross section is assumed to be a circle and the compressed fiber cross section is assumed to be a rectangle. The compressed cross sectional area is estimated using 2weff 2r u ð Þ. The normalized compressed apparent cross sectional area is found to increase to a maximum of 1.83 times the original area at 46 % nominal strain and then plateau at higher strain levels as shown in Fig. 2.5b. A more detailed study is required to understand the increase in cross sectional area. The true strain is obtained by equating the internal energy to the external work done given by eq. 2.3. The true strains are computed incrementally using trapezoidal rule to evaluate the integrals in eq. 2.3. ð V V0 ð εeff 0 σeff dεeff dV ¼ð u 0 Fdu ð2:3Þ The true stress strain response obtained by imposing a constant volume along with a linear elastic behavior with a stiffness of 2.37 GPa is shown in Fig. 2.6. The true elastic limit of 0.5 % is identified as the point at which the instantaneous stiffness is lower than the elastic stiffness of 2.37 GPa. The single fiber transverse compression experiment is modeled using a quarter symmetric finite element (FE) model shown in Fig. 2.6b in the commercial FE code LS-DYNA. The fiber is modeled as a nonlinear inelastic continuum material with a user defined constitutive model [5] UMAT. The yield stress-effective plastic strain required as input to the model is determined by subtracting the elastic portion of the true stress-strain curve (Fig. 2.6a). The UMAT force displacement predictions are much better correlation to the experimental results compared to a linear elastic behavior with stiffness of 2.37 GPa. However the model under predicts contact area compared to the experiments as shown in Fig. 2.6d. This deviation may be attributed to the increase in fiber cross sectional area observed in the experiments which is a topic for future studies. 2.4 Conclusions This paper presented the quasi-static transverse compression response of UHMWPE Dyneema SK76 ballistic fibers. The fibers exhibit nonlinear inelastic behavior under large compressive strains. The true stress strain behavior of the fiber is determined by removing the geometric nonlinearity using the measured contact area. The true elastic limit of these fibers in transverse compression is found to be 0.5 % and the Hertzian elastic limit is found to be 2 %. The numerical FE model 2 Transverse Compression Response of Ultra-High Molecular Weight Polyethylene Single Fibers 11
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