Macro to Micros Stress Conversion
Where Mini-level analysis provides the means of evaluation of individual laminate layers, Macro-level analysis provides the means of evaluating components made up of multiple laminate layers. It is based upon the assumption that not only the composite behaves as a continuum, but that the series of laminate layers acts as a homogeneous material with properties estimated based on the properties of the layer and the winding angle, and that finally, failure criteria are functions of the level of equivalent stress.
Laminate properties may be estimated by summing the layer properties (adjusted for winding angle) over all layers. For example:
Where:
ExLAM = Longitudinal modulus of elasticity of laminate
tLAM = thickness of laminate
E⊥k = Longitudinal modulus of elasticity of laminate layer k
Cik = transformation matrix orienting axes of layer k to longitudinal laminate axis
Cjk = transformation matrix orienting axes of layer k to transverse laminate axis
tk = thickness of laminate layer k
After composite properties are determined, the component stiffness parameters can be determined as though it were made of homogeneous material that is, based on component cross-sectional and composite material properties.
Normal and shear stresses can be determined from 1) forces and moments acting on the cross-sections, and 2) the cross-sectional properties themselves. These relationships can be written as:
saa = Faa / Aaa ± Mba / Sba ± Mca / Sca
sbb = Fbb / Abb ± Mab / Sab ± Mcb / Scb
scc = Fcc / Acc ± Mac / Sac ± Mbc / Sbc
tab = Fab / Aab ± Mbb / Rab
tac = Fac / Aac ± Mcc / Rac
tba = Fba / Aba ± Maa / Rba
tbc = Fbc / Abc ± Mcc / Rbc
tca = Fca / Aca ± Maa / Rca
tcb = Fcb / Acb ± Mbb / Rcb
Where:
sij = normal stress along axis i on face j
Fij = force acting along axis i on face j
Aij = area resisting force along axis i on face j
Mij = moment acting about axis i on face j
Sij = section modulus about axis i on face j
tij = shear stress along axis i on face j
Rij = torsional resistivity about axis i on face j
Using the relationships developed under macro, mini, and micro analysis, these stresses can be resolved back into local stresses within the laminate layer, and from there, back into stresses within the fiber and the matrix. From these, the failure criteria of those microscopic components, and hence, the component as a whole, can be checked.