Micro-Level Analysis - CAESAR II - Help

CAESAR II Users Guide

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Stress analysis on the "Micro" level refers to the detailed evaluation of the individual materials and boundary mechanisms comprising the composite material. In general, FRP pipe is manufactured from laminates, which are constructed from elongated fibers of a commercial grade of glass, E-glass, which are coated with a coupling agent or sizing prior to being embedded in a thermosetting plastic material, typically epoxy or polyester resin.

This means, on the micro scale, that an analytical model must be created which simulates the interface between these elements. Because the number and orientation of fibers is unknown at any given location in a FRP sample, the simplest representation of the micro-model is that of a single fiber, extending the length of the sample, embedded in a square profile of matrix.

MicroLevelAnalysisCONCEPT

Micro Level GRP Sample -- Single Fiber Embedded in Square Profile of Matrix

Evaluation of this model requires use of the material parameters of:

  1. the glass fiber

  2. the coupling agent or sizing layer normally of such microscopic proportion that it may be ignored

  3. the plastic matrix

It must be considered that these material parameters might vary for an individual material based upon tensile, compressive, or shear applications of the imposed stresses, and typical values vary significantly between the fiber and matrix (Reference 5):

Young's Modulus

Ultimate Strength

Coefficient of Thermal Expansion

Material

tensile (MPa)

tensile (MPa)

m/m/ºC

Glass Fiber

72.5 x103

1.5 x 103

5.0 x 10-6

Plastic Matrix

2.75 x 103

.07 x 103

7.0 x 10-6

The following failure modes of the composite must be similarly evaluated to:

  • failure of the fiber

  • failure of the coupling agent layer

  • failure of the matrix

  • failure of the fiber-coupling agent bond

  • failure of the coupling agent-matrix bond

Because of uncertainties about the degree to which the fiber has been coated with the coupling agent and about the nature of some of these failure modes, this evaluation is typically reduced to:

  • failure of the fiber

  • failure of the matrix

  • failure of the fiber-matrix interface

You can evaluate stresses in the individual components through finite element analysis of the strain continuity and equilibrium equations, based upon the assumption that there is a good bond between the fiber and matrix, resulting in compatible strains between the two. For normal stresses applied parallel to the glass fiber:

ef = em = saf / Ef = sam / Em

saf = sam Ef / Em

Where:

ef = Strain in the Fiber

e = Strain in the Matrix

saf = Normal Stress Parallel to Fiber, in the Fiber

Ef = Modulus of Elasticity of the Fiber

sam = Axial Normal Stress Parallel to Fiber, in the Matrix

Em = Modulus of Elasticity of the Matrix

Due to the large ratio of the modulus of elasticity of the fiber to that of the matrix, it is apparent that nearly all of the axial normal stress in the fiber-matrix composite is carried by the fiber. Exact values are (Reference 6):

saf = sL / [f + (1-f)Em/Ef]

sam = sL / [fEf/Em + (1-f)]

Where:

sL = nominal longitudinal stress across composite

f = glass content by volume

The continuity equations for the glass-matrix composite seem less complex for normal stresses perpendicular to the fibers, because the weak point of the material seems to be limited by the glass-free cross-section, shown below:

AreaOfStressIntensificationCONCEPT

Stress Intensification in Matrix Cross-Section

For this reason, it would appear that the strength of the composite would be equal to that of the matrix for stresses in this direction. In fact, its strength is less than that of the matrix due to stress intensification in the matrix caused by the irregular stress distribution in the vicinity of the stiffer glass. Because the elongation over distance D1 must be equal to that over the longer distance D2, the strain, and thus the stress at location D1 must exceed that at D2 by the ratio D2/D1. Maximum intensified transverse normal stresses in the composite are:

Where:

sb = intensified normal stress transverse to the fiber, in the composite

s^ = nominal transverse normal stress across composite

nm = Poisson's ratio of the matrix

Because of the Poisson effect, this stress produces an additional s'am equal to the following:

s'am = Vm sb

Shear stress can be allocated to the individual components again through the use of continuity equations. It would appear that the stiffer glass would resist the bulk of the shear stresses. However, unless the fibers are infinitely long, all shears must eventually pass through the matrix in order to get from fiber to fiber. Shear stress between fiber and matrix can be estimated as:

Where:

tab = intensified shear stress in composite

T = nominal shear stress across composite

Gm = shear modulus of elasticity in matrix

Gf = shear modulus of elasticity in fiber

Determination of the stresses in the fiber-matrix interface is more complex. The bonding agent has an inappreciable thickness, and thus has an indeterminate stiffness for consideration in the continuity equations. Also, the interface behaves significantly differently in shear, tension, and compression, showing virtually no effects from the latter. The state of the stress in the interface is best solved by omitting its contribution from the continuity equations, and simply considering that it carries all stresses that must be transferred from fiber to matrix.

After the stresses have been apportioned, they must be evaluated against appropriate failure criteria. The behavior of homogeneous, isotropic materials such as glass and plastic resin, under a state of multiple stresses is better understood. Failure criterion for isotropic material reduces the combined normal and shear stresses (sa, sb, sc, tab, tac, tbc) to a single stress, an equivalent stress, that can be compared to the tensile stress present at failure in a material under uniaxial loading, that is, the ultimate tensile stress, Sult.

Different theories, and different equivalent stress functions f(sa, sb, sc, tab, tac, tbc) have been proposed, with possibly the most widely accepted being the Huber-von Mises-Hencky criterion, which states that failure will occur when the equivalent stress reaches a critical value the ultimate strength of the material:

seq = Ö{1/2 [(sa - sb)2 + (sb - sc)2+ (sc - sa)2 + 6(tab2+ tac2+ tbc2)} £ Sult

This theory does not fully cover all failure modes of the fiber in that it omits reference to direction of stress, that is, tensile versus compressive. The fibers, being relatively long and thin, predominantly demonstrate buckling as their failure mode when loaded in compression.

The equivalent stress failure criterion has been corroborated, with slightly non-conservative results, by testing. Little is known about the failure mode of the adhesive interface, although empirical evidence points to a failure criterion which is more of a linear relationship between the normal and the square of the shear stresses. Failure testing of a composite material loaded only in transverse normal and shear stresses are shown in the following figure. The kink in the curve shows the transition from the matrix to the interface as the failure point.