ADMISSIBLE VALUES FOR THE INTERACTION TERM

In order to narrow the value for the interaction term of the quadratic failure criterion, we can further impose the following rationalizations:

1) We assume that there is one failure criterion for the entire range of combined stresses or strains. We can have only one value for all the strength parameters,

one of which is the interaction term. We do not wish to have different values for the interaction term in different quadrants or octants in stress or strain space.

There would be too many material parameters to determine. We must pay attention to the shape of the failure envelope in all quadrants. A forced fitting of data in one quadrant can lead to an unreasonable shape of the envelope elsewhere.

2) It is assumed that the failure envelope must be closed. The condition for a closed ellipsoid bounds the normalized interaction term between +1, and -1. This requirement ensures that there are no combined stresses or strains that would lead to infinite strength. In Figure 8.15 we show a series of failure envelopes in strain space (to be covered later in the next sub-section) having a varying interaction term between -2 and +2. Note the open surface at each end of the figure. Implicitly assumed is that all envelopes must pass through the initial strengths which act as anchor points.

3) The inclination of tangents to the failure envelopes at the four anchor points must be within the four corresponding admissible ranges cited in Figure 8.15. The slope and the inclination of the tangents are derived in the equations below in the normal stress plane where shear stress vanishes:

(8.4)

FIGURE 8.15 A RANGE OF VALUES FOR THE INTERACTION TERM OF THE

QUADRATIC FAILURE CRITERION SHOWING THE NECESSARY BOUNDS BETWEEN -1 AND +1 IF THE FAILURE ENVELOPES ARE TO BE CLOSED Our quadratic criterion is analytical: it is a closed-formed, single-valued function.

Differentiation is elementary. The inclinations at the four anchor points as functions of the normalized interaction terms are easily found:

(8.5)

By knowing the range of the inclination as a result of the failure mode interactions, we can define the range of the interaction term. The results of the first two relations of Equations 8.5 are shown in the next figure. The admissible range of the interaction term is defined by the admissible inclinations at the longitudinal tensile and compressive strengths. At the longitudinal tensile strength point, the inclination goes from -60 to -2 degrees as the interaction term goes from -1 to +1. At the longitudinal compressive strength point, the inclination goes from +2 to +60 degrees as the interaction term goes from -1 to +1. The failure mode interaction does not narrow the range of the interaction term for this ply material at these two anchor points.

FIGURE 8.16 ADMISSIBLE RANGES OF THE INTERACTION TERM AND THE INCLINATIONS OF THE TANGENTS TO THE FAILURE ENVELOPE

Next we show the range of the interaction term as defined by the transverse tensile and compressive strengths given by the last two relations in Equation 8.5. The range of the inclination goes from 1.1 to 0 degrees as the interaction term goes from -1 to zero at the transverse tensile point; it goes from 6 to 0 degrees as the interaction term goes from -1 to zero.

FIGURE 8.17 ADMISSIBLE RANGES OF THE INTERACTION TERM AND THE INCLINATIONS OF THE TANGENTS TO THE FAILURE ENVELOPE

The admissible range of the interaction term is narrowed by the range of admissible inclinations. From this figure, the range for this composite ply of T300/5208 is between 0 and -1. Positive value is not admissible by the inclination at the transverse tensile and compressive strengths, although it is admissible by the inclination at the longitudinal tensile and compressive strengths.

While the admissible range of the value for the interaction term for T300/5208 at the transverse tensile and compressive strength points is cut in half, it still covers a range from 0 to -1. It is nonetheless assuring that one value of the interaction term is adequate to satisfy four independently defined range of admissible inclinations.

Additional rationalization and/or biaxial test data will be required to more narrowly define the value for the interaction term. By repeating the process of determining the range of

interaction term of T300/5208 through the admissible range of the inclinations at each of the four anchor points, the ranges for other composite plies are shown in Figure 8.18.

FIGURE 8.18 ADMISSIBLE RANGES OF THE NORMALIZED INTERACTION TERMS Judging from the ranges of the interaction term for a vairety of materials in the figure above, the upper and lower bounds of the interaction term can be reduced from ±1 to zero and -1, respectively. A new Tsai-Wu range is more restrictive than the geometric range to ensure closed elliptic surface, as indicated in Table 8.2. It is recommended that -1/2 be used as a good approximation for all materials. Unless is otherwise specified, Tsai-Wu criterion implies that this -1/2 is used for the normalized interaction term.

FIGURE 8.19 OFF-AXIS UNIAXIAL TENSILE AND COMPRESSIVE STRENGTHS

A natural question concerns the sensitivity of the value of the interaction term in the strength prediction of composite materials under combined stresses. The off-axis uniaxial tensile and compressive strengths are insensitive to the interaction term. In fact, in the uniaxial strengths shown in Figure 9.19 show no discernible difference between the full range of values for the interaction term from -1 to +1. For a more restrictive range of interaction term between -1/2 and zero, the sensitivity will be even smaller.

The reason for the lack of sensitivity arises from the highly orthotropic strength properties of composite materials. If the failure envelope is drawn with the same scale along the fiber and its transverse, the envelope will be a highly elongated body, like a thin sausage.

The interaction term affect the ends of this elongated body more than the middle of the same body. The off-axis uniaxial tensile and compressive tests traverse near the portion of the failure envelope where the shape is not sensitive to the value of the interaction.

In fact, the sensitivity of the interaction term depends on the ply material, and the externally imposed stresses. The insensitivity of the interaction term on the 2-dimensional pressure on a 0-degree specimen, and a uniaxial compression on a 45-degree

specimens for two ply materials for the entire range of admissible value for the interaction term is shown in the figure below. Horizontal lines mean complete insensitivity of the interaction term.

FIGURE 8.20 PLANE (2-D) PRESSURE AND UNIAXIAL COMPRESSIVE STRENGTHS OF T300/5208 AND E-GLASS/EPOXY COMPOSITE MATERIALS FOR A FULL RANGE OF THE INTERACTION TERM