Mini-Level Analysis - CAESAR II - Help

CAESAR II Users Guide (2019 Service Pack 1)

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CAESAR II Version
11.0 (2019)

Mini-Level Analysis Fiber Distribution Models

Although feasible in concept, micro level analysis is not feasible in practice. This is due to the uncertainty of the arrangement of the glass in the composite the thousands of fibers that might be randomly distributed, semi-randomly oriented, although primarily in a parallel pattern, and of randomly varying lengths. This condition indicates that a sample can truly be evaluated only on a statistical basis, thus rendering detailed finite element analysis inappropriate.

For mini-level analysis, a laminate layer is considered to act as a continuous hence the common reference to this method as the "continuum" method, material, with material properties and failure modes estimated by integrating them over the assumed cross-sectional distribution, which is, averaging. The assumption regarding the distribution of the fibers can have a marked effect on the determination of the material parameters. Two of the most commonly postulated distributions are the square and the hexagonal, with the latter generally considered as being a better representation of randomly distributed fibers.

The stress-strain relationships, for those sections evaluated as continua, can be written as:

eaa = saa/EL - (nL/EL)sbb - (nL/EL)scc

ebb = -(nL/EL)saa + sbb/ET - (nT/ET)scc

ecc = -(nL/EL)saa - (nT/ET)sbb + scc/ET

eab = tab / 2 GL

ebc = tbc / 2 GT

eac = tac / 2 GL

Where:

eij = strain along direction i on face j

sij, tab = stress (normal, shear) along direction i on face j

EL = modulus of elasticity of laminate layer in longitudinal direction

nL = Poisson’s ratio of laminate layer in longitudinal direction

ET = modulus of elasticity of laminate layer in transverse direction

nT = Poisson’s ratio of laminate layer in transverse direction

GL = shear modulus of elasticity of laminate layer in longitudinal direction

GT = shear modulus of elasticity of laminate layer in transverse direction

These relationships require that four modules of elasticity, EL, ET, GL, and GT, and two Poisson’s ratios, nL and n, be evaluated for the continuum. Extensive research (References 4 - 10) has been done to estimate these parameters. There is general consensus that the longitudinal terms can be explicitly calculated; for cases where the fibers are significantly stiffer than the matrix, they are:

EL = EF f + EM(1 - f)

GL = GM + f/ [ 1 / (GF - GM) + (1 - f) / (2GM)]

nL = nFf + nM(1 - f)

You cannot calculate parameters in the transverse direction. You can only calculate the upper and lower bounds. Correlations with empirical results have yielded approximations (Reference 5 and 6):

ET = [EM(1+0.85f2) / {(1-nM2)[(1-f)1.25 + f(EM/EF)/(1-nM2)]}

GT = GM (1 + 0.6Öf) / [(1 - f)1.25 + f (GM/GF)]

nT = nL (EL / ET)

Use of these parameters permits the development of the homogeneous material models that facilitate the calculation of longitudinal and transverse stresses acting on a laminate layer. The resulting stresses can be allocated to the individual fibers and matrix using relationships developed during the micro analysis.