# Shear Deformation

The parallel-plate geometry imposes a torsional deformation; the strain on the material varies from zero at the center to a maximum at the rim (edge) of the moving plate. Therefore, the deformation geometry is defined by plate radius R and the gap h between the plates such that the maximum shear strain on a test specimen arising from an angular displacement ® relative to its initial state of rest is

R

Jr = © (1)

This is not the case for the cone-and-plate geometry where, for cone angles 3< 6°, the variation in shear strain across the plate radius is less than 1 % [12] such that the following approximation holds

The mathematical relationship between j and joint movement can be de­rived following the schematic diagram in Fig. 3. A unit volume originally at rest, represented by the solid line, is deformed by shearing its top surface re­sulting in a change of shape denoted by the dotted line. Noting that the bottom surface remains fixed, shear strain is the ratio of the change in shape defined by the displacement A to the height of the unit volume h

j = A/h (3)

If joint movement can be defined as m X 100 %, then geometrically according to the Pythagorean theorem

[(1+m)h]2 = h2 + A2 = h2 + (jh)2 (4)

(1 + m)2h2 = (1 + j2)h2 (5)

 FIG. 3—Schematic diagram for a unit volume in shear deformation.

m = Vl + у2 – 1 (6)

For example, a 15 % movement corresponds to a shear strain of 56.8 %.

Controlled-Strain Cyclic Deformation

A method for evaluating the fatigue properties of structural silicone glazing materials was recently proposed by Carbary et al. [13] to simulate shear defor­mation typical in curtainwall designs due to daily thermal movement differen­tials. The proposed test method subjected specimens, prepared according to the ASTM C1135-00 “Standard Test Method for Determining Tensile Adhesion Properties of Structural Sealants” [14], to 36 500 cycles of 15 % movement at a frequency of five cycles per minute. A total of nine (one – and two-part) silicone sealants representing movement capabilities from 12.5 to 50 % were investi­gated. The results from ASTM C1135 tensile adhesion testing before and after the fatigue test revealed a modulus reduction for each sealant.

Figure 4 reproduces some of the results from Ref [13] in terms of the 25 % secant modulus. In general, the trend suggested that sealants with a higher initial modulus (t = 0) exhibited a larger modulus reduction (approaching 40 %) compared to the lower modulus elastomers (~20 % reduction). Nevertheless, it was evident that this approach, which produced only two data points from each experiment, provided no degrees of freedom in proposing an appropriate fa­tigue mechanism that was not linear with time (or number of fatigue cycles).

A test protocol was set up for the ARES controlled-strain rheometer to follow the deformation history described in Ref [13]. From Eq 6, a shear strain of 56.8 % using a dynamic frequency of 0.083 Hz was defined. For each sealant listed in Table 1, more than 104 data points of the stress response, starting approximately 12 s after starting the test, were collected over more than five days of cycling. Figure 5 shows the results obtained using the parallel-plate geometry where, for the purpose of clarity, only every 300th data point was plotted.

All four sealants exhibited an overall shear stress reduction ranging, as listed in Table 2, from 10 % for the one-part alkoxy Sealant B to 17 % for the one-part acetoxy Sealant A. This reduction was consistent with the 25 % secant modulus data plotted in Fig. 4; however, the results in Fig. 5 revealed more

 Si Sealant Movement, %

 180

 + 2-part alkoxy 12.5 □ 2-part alkoxy 12.5 V 2-part alkoxy 20 1-part alkoxy 25 0 1-part oxime 25 Д 1-part oxime 25 x 1-part alkoxy 50

 FIG. 4—Effect of cycling at 15 % shear movement on the 25 % secant modulus from ASTM C1135 testing reported in Ref (13).

Number of cycles at 5 cycles per minute

 FIG. 5—Effect of cycling at 56.8 % torsional strain (15 % movement) on the stress response for four cured silicone sealants (Table 1) at 25 ° C. Data edited for clarity by plotting every 300th point.

TABLE 2—Sealant response to 0.083 Hz sinusoidal deformation of 56.8 % shear strain at 25 ° C.

___________________________ Shear stress (MPa)____________________________

Silicone Minimum 1 – Minimum/Initial, Final 1 – Minimum/Final,

 Sealant Initial (time, h) % (5.1 days) % Parallel plate A 0.167 0.139 (23) 17 0.142 1.5 B 0.200 0.181 (18) 9.7 0.184 1.9 C 0.283 0.245 (122) 14 0.245 N/A D 0.375 0.325 (20) 13 0.330 1.6 Cone-and-plate B 0.179 0.164 (15) 8.5 0.167 1.6 D 0.271 0.236 (8) 13 0.249 5.2

aspects to the stress response. First, the majority of the stress/modulus reduc­tion occurred within the first 24 hours of cycling at 57 % shear strain. This strain-induced stress-softening phenomenon with successive cycling is typical in filled systems and has been referred to as the Mullins effect [15]. This effect was attributed to polymer chains, having reached its limit of extensibility, de­taching from the filler surface or slipping on the filler surface [16]. The Mullins effect has been closely related to the mechanical fatigue of elastomers and, consequently, has received a lot of attention in addressing the durability and service life of these systems. To be consistent with the experimental protocol in Ref [13], the test specimens were not preconditioned by imposing a strain greater than 57 % to remove the Mullins effect. Therefore, the stress-softening response in Fig. 5 was not unexpected, noting that shear is the least damaging such that the Mullins effect is small relative to other types of deformation [17].

More noteworthy was the subsequent stress recovery observed for the re­mainder of the test period in three of the four sealants, which is listed in the last column of Table 2. Relative averages for the standard deviation (0.006 %) and range (0.13 %) of the actual strain data over each five-day test period could not account for a significant contribution to the observed stress recovery. To determine if this response was a consequence of torsional (nonuniform) defor­mation, two of the sealants were tested at uniform shear strain using the cone- and-plate geometry where similar trends were observed as well (Table 1 and Fig. 6). Hence, it would appear that some sealants are able to exhibit a modest recovery during the remaining four days of cyclic deformation. While noting that the magnitude of the Mullins effect depends in part on the sealant formu­lation, White and Hunston [18] also observed recovery from the Mullins effect given sufficient periods of time between repeated loading cycles. For future considerations, it may be revealing to investigate both the effects of cycling frequency and strain amplitude on the rate of recovery from the Mullins effect.

Another issue of concern was the nonlinear mechanical response of seal­ants. Figure 7 shows the stress-strain profile for two sealants tested using the cone-and-plate geometry and a dynamic frequency of 0.083 Hz. A deviation from a linear correspondence (given by each straight line) was evident well

 FIG. 7—Stress-strain response in dynamic shear deformation for two cured sealants at 25 ° C.

 FIG. 8—Waveform of the stress response relative to the input sinusoidal strain of Seal­ant D using the cone-and-plate geometry.

 FIG. 7—Stress-strain response in dynamic shear deformation for two cured sealants at 25 ° C.

below the 57 % shear strain applied during the cycling testing. A sinusoidal waveform cannot be expected from a nonlinear response to a sinusoidal input. An example is shown in Fig. 8 for Sealant D, where the input and output wave­forms were monitored using the RheoChart software accompanying the Wave­form & Fast Data Sampling Option with the ARES rheometer. A distorted out­put waveform was evident relative to the input sinusoidal waveform. Therefore, no attempt was made to decompose the dynamic response into its elastic (stor­age) and viscous (loss) modulus components [19].