Can we determine shear modulus from uniaxial tests of [45] and [±45]?

FIGURE 4.17 POISSON'S RATIOS OF [152/60]s LAMINATES

Sample 6: Can we determine shear modulus from uniaxial tests of [45] and [±45]?

Yes, we need to know the transformation relation of compliance at 45 degrees as follows:

(4.24) We can establish the following relations for T300/5208 and E-glass/epoxy composites.

The [±45] laminates are calculated for each value of the shear modulus for the same materials. There is no closed form relation for laminates.

FIGURE 4.18 FIGURES TO BACK-CALCULATE SHEAR MODULUS FOR T300/5208 AND E-GLASS/EPOXY COMPOSITES FROM [45] AND [±45] TEST SPECIMENS From measured Young's modulus of [45] and [±45], we can use the figures above to find the implied shear modulus. The solid lines with arrows in each figure represent the

"perfect" measurements that would recover the original shear modulus of the ply material.

4.7 TRANSFORMATION AND INVARIANTS OF IN-PLANE MODULI

The transformation relations for the in-plane stiffness and compliance matrices follow precisely those for the ply stiffness and compliance, respectively; see Section 3, Ply Stiffness. The invariants associated with the transformation should be maintained as we go from plies to laminates. This is shown in the figures below for the stiffness and compliance.

FIGURE 4.19 INVARIANCE OF UNIDIRECTIONAL AND ANGLE-PLY LAMINATES

FIGURE 4.20 COMPLIANCE INVARIANCE OF UNIDIRECTIONAL AND ANGLE-PLY LAMINATES

The trace of the compliance does not go from the unidirectional ply to that of a laminate because, in laminated plate theory, strain, not stress, is assumed to be constant.

4.8 PLY STRESS AND PLY STRAIN

It is useful to examine the ply stress and ply strain defined earlier in this section. Figures 3.2 and 3.3 on page 3-3 for the ply stiffness transformation are modified here to suit the ply-by-ply stress analysis of a symmetric laminate under in-plane loading.

FIGURE 4.21 DETERMINATION OF PLY STRESS AND STRAIN FROM LAMINATE STRESS

The actual formulas for transformation and stress-strain relation are shown in Figure 4.22 below:

FIGURE 4.22 DETERMINATION OF PLY STRESS AND STRAIN FROM LAMINATE STRESS

Equilibrium check is one way of verifying the ply-by-ply stress analysis. The following relations should be used:

(4.25) This is demonstrated by an example given below where a [ /4] quasi-isotropic laminate is subjected to a combined in-plane stress of {20,0,40}. The ply strains and ply stress in the laminate axis are shown in the table below. Note that ply strains are equal to laminate strains, as required by laminated plate theory. Ply stresses are different from laminate stress, as expected. The average of the ply stress must be equal to the laminate stress, as required by equilibrium stated in the first part of Equation 4.25.

One important point should be made about the motivation for the ply-by-ply stress analysis of a laminate. Being tensors, both stress and strain components are dependent on the reference coordinates. The components vary. It is difficult to say if the stress or strain is high or low, or safe or unsafe. For scalars, this is easy to do. If the temperature

is 100°C, water boils. The most effective way of assessing the magnitudes of stress and strain is by their invariants, which, by definition, are scalars. When we discuss failure criteria, we will make a strong bid for using the quadratic failure criterion, which is a scalar criterion. The maximum stress and maximum strain criteria are not scalar criteria.

Because of this and other basic flaws, they are not recommended.

TABLE 4.3 PLY-BY-PLY STRAIN AND STRESS VARIATIONS IN A LAMINATE

This numerical values are plotted in the following figures, where the ply layup, ply strains, and ply stresses are shown, respectively. The laminate stresses are also shown as a heavy vertical lines

FIGURE 4.23 PLY-BY-PLY STRAIN AND STRESS DISTRIBUTIONS OF A LAMINATE 4.9 RESIDUAL STRESSES

Organic and inorganic matrix composites will have very complicated residual stresses after processing or curing. On the micromechanical level, processing or curing stresses are caused by the volumetric contraction of the matrix, the differential thermal contraction between the matrix and the fiber after cooldown, and non-uniform consolidation or solidification. For organic matrix composites, moisture is absorbed which introduces additional residual stresses. The effects of these stresses are difficult to assess and cannot be measured directly. The empirically measured ply strengths are very much affected by the residual stresses. The effects are, in fact, reflected in the measured strengths. Until reliable predictions of strength based on micromechanics become available, we will back-calculate residual stresses from the temperature-dependent strength data.

Another set of residual stresses originates from the macromechanical or laminate level.

Because composite plies are anisotropic, the thermal expansion or contraction in the longitudinal direction is much less than that in the transverse direction. This differential contraction after cooldown, and expansion after moisture absorption will give rise to macromechanical residual stresses among plies in a multidirectional laminate. Using laminated plate theory, these stresses are relatively easy to calculate.

We are only concerned with the macromechanical residual stresses in this section. We assume that temperatures before and after curing and moisture absorbed after curing remain uniform across the laminate thickness. We can extend the theory to deal with a linearly varying temperature across a symmetric plate as a special case.

The stress-free expansion of a unidirectional ply is shown in the figure below. The free on-axis expansions of a ply are:

FIGURE 4.24 STRESS-FREE EXPANSIONS OF A UNIDIRECTIONAL PLY. THE

REFERENCE STATE IS UNCURED PLIES AT CURE TEMPERATURE; THE EXPANDED STATE IS BASED ON DIFFERENCES IN TEMPERATURE AND MOISTURE CONTENT AFTER CURING

Strengths of unidirectional composites are commonly measured after cooldown and an anticipated exposure to moisture over a long period of time. While temperature is usually uniform within the composite, the moisture is almost always non-uniform. The slow diffusion of moisture is responsible for this non-uniformity. The measured strength or the corresponding ultimate strain is depicted in the figure below. The strain from the original stress-free state at the cure temperature must be the sum of the free expansion and the measured mechanically applied strain at room temperature.

FIGURE 4.25 MEASURED ULTIMATE STRAINS AFTER FREE HYGROTHERMAL EXPANSION OR CONTRACTION

4.10 RESIDUAL STRAINS AFTER CURING

The curing of a multidirectional laminate induces macromechanical curing stresses. This is shown in the figure below. Although the laminate in this figure is a simple cross-ply, the principle is applicable to all laminates. The mathematical formulation in this section is approximate because the process of curing an organic matrix is in general time-dependent and nonlinear. We use only time-intime-dependent, linear theory. One simple way to compensate for this deficiency is to use the stress-free temperature in place of the actual cure temperature. We have found that the stress-free temperature can be as much as 50 degrees C below the cure temperature. The simplest method of determining

the stress-free temperature is to observe the elevated temperature at which a warped unsymmetric laminate becomes flat.

FIGURE 4.26 RELATION BETWEEN NONMECHANICAL, RESIDUAL, AND FREE EXPANSION STRAINS AS DEFINED BY THE EQUATION IN THE FIGURE.

ALL STRAINS ARE IN-PLANE AND RELATIVE TO THE MATERIAL AXES.

THE LAMINATE IS SYMMETRIC.

In order for the strain components to be additive in the figures above, they all must be in the on- or off-symmetry axes of the plies. The nonmechanical strain is the laminate strain measured from the stress-free state. The residual strain is simply the difference between the nonmechanical strain and the free expansion strain. We will now derive these strains from laminated plate theory.

The nonmechanical stresses are derived from the traction-free nonmechanical strains given in Figure 4.24 above:

(4.26) For a on-axis orthotropic material, there is no nonmechanical shear stress.

The transformed nonmechanical stress from the ply- to the laminate-axis:

(4.27) We can now derive the nonmechanical stress components:

(4.28) If we limit this calculation to symmetric laminates, the nonmechanical in-plane strains shown in Figure 4.26 above are:

(4.29) The strains here are in the on- or material symmetry-axis, not in the laminate-axis.

For cross-ply laminates, we have the following unique relations, where nonmechanical stresses are hydrostatic in the plane of the laminate:

(4.30) To find the resulting strains, we need the compliances of both laminates. We take T300/5208 material, and assume that we have a -100°K temperature difference. Since the compliances for the laminates are equal, the resulting thermal strains are also hydrostatic, as shown in the equation below. The laminates are thermally isotropic.

(4.31) The quasi-isotropy for hygrothermal properties is simpler to attain than the elastic moduli.

The reason is that the hygrothermal are second rank tensors, and the latter, fourth rank tensors. Thus [0/90] are isotropic for hygrothermal properties, while equal ply orientation of [π/3], [π/4], and higher-order laminates are required to attain elastic isotropy. A cross-ply laminate of [0/90] is not isotropic elastically.

The residual strain is a function of temperature difference (usually negative) and moisture concentration. If both are zero, the residual strain is of course zero. If we have a unidirectional composite (without lamination), the residual strain is also zero.

If the operating temperature is equal to the cure temperature, there will be no residual strain due to curing. In this case, the residual strain due to moisture will be a linear function of the moisture concentration. If moisture concentration is zero, the residual strain will be a linear function of temperature difference. If both temperature and moisture are not zero, the residual strain will be nonlinear. This nonlinearity is important if we wish to calculate the "self-destruct" temperature or moisture level.

4.11 EXPANSION COEFFICIENTS

The effective in-plane expansion coefficients are formulated by setting either temperature or moisture at zero; i.e., the free expansion is computed by assuming that it is either due to temperature or moisture, but not both. Having the hygrothermal expansion coefficients, we can calculate the expansion strains as follows:

(4.32) The thermal expansion coefficients can be obtained by integrating the nonmechanical stress using the method of the last sub-section. Free thermal expansions per degrees are by definition thermal expansion coefficients:

(4.33) The moisture expansion coefficients can be similarly integrated using the same method of the last sub-section where free moisture expansions are replaced by moisture expansion coefficients:

(4.34)

These laminate expansions can be positive, zero, or negative, and can induce shear by having a nonzero "6" component. By properly designing the laminate layup, unique expansion behavior is possible.

Hygrothermal expansion coefficients for typical composite materials are listed in the table below:

TABLE 4.4 TYPICAL HYGROTHERMAL EXPANSION COEFFICIENTS

4.12 SAMPLE CALCULATIONS IN EXPANSIONS

Sample 1: Find the free hygrothermal expansions of CFRP T300/5208 at room