Abstract

Described here is the design and development of a computer-controlled device capable of measuring the finite strain thermomechanical behavior of a general class of polymeric materials including elastomers and biological soft tissues. The utility of this device for thermoelastic and thermophysical investigations is demonstrated by the measurement of the in-plane stress-stretch response and in-plane and out-of-plane components of thermal diffusivity of neoprene rubber undergoing finite deformations. ͓S0021-8936͑00͒01603-2͔ Introduction The thermomechanical behavior of elastomeric, and specifically rubber-like, materials has generated considerable interest in the mechanics community for close to two centuries. With the advent of laser, ultrasound, and microwave-based medical devices, a similar interest has recently arisen in the thermomechanical behavior of soft tissues ͓͑1͔͒. A general thermomechanical analysis of these material behaviors requires, of course, both multiaxial thermoelastic and thermophysical data. Likewise, formulation of nonlinear constitutive relations necessitates appropriate theoretical frameworks to design and interpret the requisite experiments ͓͑2͔͒. For example, theory reveals that a complete description of reversible finite strain thermomechanical behavior requires identification of two independent constitutive functions ͓͑3͔͒, the Helmholtz potential ϭ ͑ C,T ͒, and the referential heat flux vector where C (ϭF T •F) is the right Cauchy-Green deformation tensor, F (ϭ‫ץ‬x/‫ץ‬X) the deformation gradient tensor, T the temperature, " 0 T (ϭ‫ץ‬T/‫ץ‬X) the referential temperature gradient, and x and X the position of a material particle in the current and reference configurations, respectively. Stress-strain-temperature relations result from derivatives of with respect to C. Although a number of functional forms for have been suggested ͓͑4-7͔͒, a widely accepted form remains elusive due, in large part, to the continuing lack of multiaxial thermoelastic data. That is, most investigators have focused on describing the available uniaxial data ͓͑8-11͔͒ and hence the peculiar one-dimensional Gough-Joule and thermoelastic inversion effects. Recently, however, Ogden ͓7͔ proposed a method for finding as a function of biaxial stretches and temperature, whereas Humphrey and Rajagopal ͓12͔ showed that in-plane biaxial tests allow measurement of thermoelastic response functions ͑e.g., ‫ץ‬ /‫ץ‬I C where I C ϭtr C͒ similar to the isothermal results of Rivlin and Saunders ͓13͔. Both cases require measuring biaxial stress and stretch at multiple temperatures. Alternatively, there has been little attention to the possible finite strain dependence of the heat flux. Rather, most reports assume Fourier conduction q(x,T)ϭϪk(T)"T where q(x,T) is the spatial heat flux, "T (ϭ‫ץ‬T/‫ץ‬x) the spatial gradient of temperature, and k(T) the scalar ͑i.e., isotropic͒ spatial thermal conductivity ͓͑14͔͒. For finite strains, referring the conductivity tensor to the reference configuration simplifies material symmetry considerations, and thereby facilitates the formulation of general constitutive relations. Thus, note that qϭ(1/J)F•q o , where Jϭdet F. Moreover, a generalized Fourier conduction has the form where K(C,T) is the referential thermal conductivity tensor. Of the tractable finite strain tests ͑e.g., combined extension and torsion of a cylinder, membrane inflation, etc.͒, the in-plane biaxial extension of a thin rectangular sheet ͓͑13,15͔͒ is also convenient for thermophysical testing. Not only is the resulting strain field homogeneous in the central region and the state of stress planar, thin specimens facilitate isothermal testing and measurement of thermal diffusivity. For example, Doss and Wright ͓16͔ recently demonstrated that the transient flash diffusivity method ͓͑17͔͒ may be extended to measure the diagonal components of the spatial thermal diffusivity tensor ␣ of thin sheets of stiff polyvinyl chloride ͑PVC͒. Specifically, the flash diffusivity method yields ␣ via the spatial energy equation which, in the absence of stress power and volumetric heat addition, is where t is time. For materials in mechanical equilibrium, the convective terms within the material are negligible, of course, and the total derivative on the left-hand side of Eq. ͑4͒ reduces to a partial derivative with respect to time. Appendix A contains an outline of the one-dimensional solution of Eq. ͑4͒ that Parker et al. ͓17͔ employed in the initial description of the flash method. Here, Eq. ͑4͒ is solved using a finite difference formulation as part of a Marquardt parameter estimation algorithm for both the traditional ͑one-dimensional͒ flash method ͑measuring the out-of-plane component ␣ 33 only͒ and the extended flash method ͑measuring the three diagonal components of ␣͒. There is, therefore, a clear need for a multiaxial thermomechanical test system that can exploit the available theoretical results. This paper describes the design and construction of a new device capable of both in-plane biaxial thermoelastic testing and measurement of the orthogonal components of ␣. Illustrative data are presented for neoprene subject to equibiaxial stretch ratios ϵ 1 ϭ 2 ͑ i ϭl i /L i , no sum on i, with l i and L i being current and reference lengths, respectively͒ with ͓1,1.5͔, at temperature levels of 21 and 41°C. Experimental System Figure 1 is a schema of the overall optical-thermomechanical system. The system consists of five subsystems, one each for ͑a͒ biaxial loading, ͑b͒ in-plane strain measurement, ͑c͒ environmental control, ͑d͒ flash illumination, and ͑e͒ point-wise temperature measurement. Biaxial Loading System. The load frame is machined from one piece of mild steel to outer dimensions of 45.7ϫ45.7 cm with a 2.54 cm square cross section. The outer and inner surfaces are ground to ensure that opposing sides are flat and parallel. The frame is mounted on an optical table using a standard 2.54-cm diameter, 15.2-cm long support rod at each corner. A pair of horizontal through-holes ͑1.27-cm diameter and 4.45 cm apart͒ are centered on each side of the frame ͑Fig. 2͑a͒͒ such that the axes of any pair of holes are collinear with those of the opposing pair and perpendicular to the other two pairs. Each of the eight holes is fitted with a 1.91-cm-long linear recirculating ball bearing specifically designed for linear travel of shafts with minimal transverse play. Nearly uniformly distributed in-plane biaxial forces are applied to the square specimen through four load carriages ͑Fig. 2͑b͒͒. Each edge of the specimen is attached to a single carriage using Kevlar thread ͑or silk suture for biological tissues͒. The load carriages consist of two parallel 0.635-cm-diameter solid, hardened stainless steel rods. Each rod is supported in one of the aforementioned through-holes by one of the linear bearings in the load frame and a neoprene O-ring sandwiched between two Rulon bearings, which form a watertight seal for the environmental chamber. Proper alignment of this chamber within the load frame ensures free and smooth translation of the load carriages. The outside end of each pair of rods is attached to a cross bar that separates the shafts and facilitates the application of axial forces. The inside end of each load carriage consists of three separate aluminum pieces: a cross bar that connects the two rods, a coupling bar, and a T-bar that has a series of equally spaced holes ͑0.75 cm apart͒ that serve as rigging points for the specimen loading threads. Two of the load carriages, one on each axis, have watertight and temperature compensated (T͓Ϫ29,93͔°C) load cells mounted between the cross bar and coupling bar. The last component of the loading subsystem is the mechanism for inducing axial load. Each of the four carriage assemblies is independently loaded via a 1-mm lead ball-screw driven by a stepper motor. A ball-nut attached to the outer cross bar of each load carriage converts the rotation of the ball-screw into linear displacement of the carriage assembly. The stepper motors are individually controlled by a four-axis indexer card in the Pentium personal computer ͑PC͒. Such a drive system allows implementation of fully automated stretching protocols. Strain Measurement. In-plane finite strains are measured optically by tracking the position histories of four small, contrasting markers that are affixed to the bottom surface of the specimen ͑Fig. 2͑b͒͒. This approach has been described previously ͓͑18͔͒, and is sufficient because of the homogeneity of the strain field in the central region, as confirmed via pilot experiments as well as by finite element analysis. For example, finite element analysis ͑ABAQUS͒, assuming a Mooney-Rivlin material response, revealed that the strain field is essentially homogeneous and extensional ͑Ͻ 5 percent shear͒ in the central sixteenth of the planar area of a square elastomeric specimen ͓͑19͔͒ that is loaded by five equidistant point loads at each edge. A CCD video camera, framegrabber board in the PC, and custom software track the position of each of the four markers at the 30 Hz frame rate. The software algorithm, based on Downs et al. ͓20͔, uses a correlation method to locate the markers first in a ''coarse'' and then in a ''fine'' search region. The marker positions serve as input to a bilinear isoparametric interpolation algorithm that provides the components of F in the central region at each configuration ͑see Appendix B͒. The components of F provide information for feedback control ͑described below͒ of the thermoelastic tests at thermal equilibrium. In thermophysical tests, the components of F are reg- Transactions of the ASME istered prior to the flash illumination ͑i.e., at mechanical equilibrium͒, which only slightly perturbs the strain field. Environmental Chamber. The environmental chamber allows testing of the sample in air or liquid at nearly constant temperatures ͑Ϯ1°C͒. The chamber is constructed from a 1.27-cmthick polycarbonate sheet and sealed with silicone adhesive to prevent leakage of the solution. With outside dimensions 40.0 ϫ40.0ϫ7.62 cm, the polycarbonate shell fits inside of the load frame ͑Fig. 2͑a͒͒. Though not shown for clarity, top and bottom plates are screwed onto the polycarbonate shell and silicone rubber gaskets form seals between the plates and shell. The top plate of the chamber has a central port that allows the flash lamp ͑described below͒ to be placed in close proximity to the specimen, thereby ensuring that enough energy is absorbed by the sample to obtain a desirable temperature rise on its bottom surface. The separation between the lamp and an approximately 2-mm-thick specimen may be adjusted between 1.0 and 3.0 cm. In addition, the top plate allows access by the thermocouple probe for measurement of the temperature field on the bottom face of the sample. The bottom plate has a central 96ϫ96ϫ1.5 mm glass window that allows the camera to view the aforementioned markers for strain measurement. Additionally, the bottom plate has drains to remove liquid from the chamber. The chamber can be maintained between room temperature and 90°C by a 750 W, 1.27-cm-diameter, 30.5-cm-long submersible heater that is mounted in the chamber ͑Fig. 2͑a͒͒ and modulated by a thermostatic controller. Temperature Measurement and Flash System. The principle of the flash technique for measuring thermal diffusivity is described elsewhere ͓͑21-23,17,24͔͒. Briefly, a flash system consists of a radiant energy source ͑e.g., a flash lamp or pulsed laser͒ capable of delivering a short burst that heats one face of the specimen, and a probe to measure the associated temperature history on the opposite face. Here, a linear xenon flashtube, mounted in an aluminum reflector, illuminates the top face of the specimen. When the in-plane components of ␣ are to be measured, a 50 ϫ50 mm aperture plate with a central 20ϫ20 mm opening is mounted between the source and the specimen; the aperture is removed for measurement of ␣ 33 alone. Three 0.25-mm-diameter E-type thermocouple probes are mounted in an aluminum bar and form the apexes of a right isosceles triangle having two 15-mm sides, and when aligned with the aperture, they measure the temperature at the center of the projection of the lighted area and at two points outside this projection along the two in-plane axes. For measurements of ␣ 33 alone, only the central thermocouple output need be used. A small amount of high thermal conductivity silicone paste is used to insure good contact between the thermocouple probe and the specimen surface. The fixture is attached, through the port in the top plate of the environmental chamber, to a micrometer head that can raise or lower the thermocouples as required. A second CCD camera and mirror ͑not shown͒ monitors the contact between the specimen and the thermocouples. Data Acquisition and Control. Eight independent channels of information are recorded simultaneously using an analog-todigital ͑A/D͒ conversion board in the PC. The A/D board has high-gain amplification and a cold junction compensation circuit, specifically designed for acquisition of thermocouple data, recorded on one of the eight channels. The E-type thermocouples occupy three channels, whereas two T-type thermocouples, one at the glass window covering the flashbulb to mark the flash event on the data file and another above the specimen to record the ambient temperature, occupy two channels. The two load cells are connected to a signal conditioner, the output of which occupies the last two channels. The strategy to control the biaxial finite deformation involves actuating the two opposing motors in one stretching direction at a constant velocity and varying the velocity of the two motors in the orthogonal direction so that the measured stretch in that direction is within a small error ⑀ of the desired stretch. Hence, let d be the desired stretch ͑either 1 or 2 ͒, a be the measured stretch, and ␦ the difference between a and d . If ͉␦͉р⑀, then the velocities of the two motors on that axis are set to zero. If ͉␦͉Ͼ⑀, however, then the motor velocities must be adjusted to bring a back to within ⑀ of d . The velocities of the controlled motors are adjusted proportionally to ␦ using ϭG␦, where G is a suitably valued parameter determined by trial and error during preliminary tests until the control of the specimen is acceptable. If G is too small, the stepper motors move continuously in the same direction, indicating that is too low and possibly not keeping the stretches within ⑀. Conversely, if G is too large, the motors oscillate, indicating an overshoot of d , thus requiring the motors to reverse. A suitable value of G between these two extremes should thus be selected. Note that if ␦Ͼ0, then a Ͼ d and the direction of rotation of the motors must be such that a is decreased; if ␦ Ͻ0 then the specimen must be stretched more. Once the controlling velocity is determined, its value is sent to the motor controller card. The control cycle is repeated throughout the experiment at 30 Hz by calculating the new stretch ratios, determining the controlling velocity, and sending the new velocity to the motor controller card. Specimen Preparation. Thermoelastic tests were performed on specimens measuring 50-mm square cut from 1.6-mm-thick precast sheets of high-grade neoprene rubber with a Shore A hardness value of 35-45 ͑McMaster Carr͒. Outside of the biaxial device, each edge of the specimen was sewn to a T-bar using a Journal of Applied Mechanics SEPTEMBER 2000, Vol. 67 Õ 467 0.254-mm-diameter sewing needle and 0.2-mm-diameter Kevlar thread. Six holes spaced about 3.75 mm apart and 10 mm from the edge of the specimen were used on each edge. Four white 200-m spots of titanium white acrylic paint were placed in the central 5ϫ5 mm square ͑i.e., 1/100th of the planar area͒ of the specimen on the bottom face of the black specimen. The T-bars were then attached to the coupling bars ͑Fig. 2͑b͒͒. Prior to thermoelastic testing, each specimen was systematically preconditioned, thermally and mechanically: each neoprene sample was held at an in-plane equibiaxial stretch ratio of ϭ1.45 at 41°C for 24 hours. The reduced range of temperatures examined in these first measurements on neoprene, as compared with the 90°C capability of the chamber, are the result of the reduced tear resistance of neoprene at elevated temperatures. Illustrative Results Stress-Strain Response. For homogeneous principal extensions Fϭdiag͓ 1 , 2 , 3 ͔ and Jϭdet Fϭ 1 2 3 ϭ 0 / ͑where 0 and are the reference and current mass densities, respectively͒, hence 3 can be determined at each temperature, given ϭ(T). Data obtained from Anter Laboratories ͑Pittsburgh, PA͒ on neoprene samples tested at T ͓20,60͔°C suggest that, to first order, where 0 ϭ1.317 g/cm 3 , ␤ϭ3.915ϫ10 Ϫ4 (°C) Ϫ1 , and T 0 ϭ20°C is the reference temperature at which 0 is measured. Thermal Diffusivity. Two types of tests illustrate the ability of the device to measure thermal diffusivity of specimens subjected to finite in-plane deformation: measurements of ͑a͒ ␣ 33 for three specimen thicknesses and multiple equibiaxial stretches and ͑b͒ the diagonal components of ␣ as a function of finite equibiaxial deformation at room temperature. First, ␣ 33 was measured for three neoprene specimens of nominal thicknesses of 1.6, 2.4, and 3.2 mm. Following preconditioning and then registration of the unloaded reference configuration ͑i.e., recording the marker positions͒ at T 0 Ϸ21°C ͑i.e., ambient room temperature͒, the top surface of each specimen was subjected to a series of five pulses from the flash lamp, each separated by ten minutes to allow the specimen to regain thermal equilibrium. Since the transient temperature rise on the bottom surface due to each flash was about 3°C and of similar magnitude on the top face, after a brief initial transient ͑10 ms͒, data were collected at nearly mechanical and thermal equilibrium. Here, the bottom surface temperature history was measured directly and ␣ 33 was calculated using the Marquardt parameter estimation algorithm coupled with a finite difference solution of Eq. ͑4͒ outlined in Appendix A. A second set of tests measured the diagonal components of ␣ for three specimens with a nominal thickness of 2.4 mm. Again, following preconditioning, each specimen was tested at two equibiaxial stretch states of approximately ϭ1.03 and 1.52. A minimum of five flash tests were performed at each of the two deformation states. At the smaller deformation (ϭ1.03), ␣ 33 is found to be 0.116 mm 2 /s, which is within 2.4 percent of the value measured with the one-dimensional test and within the standard deviation of both tests. The in-plane values are found to be 0.151 and 0.153 mm 2 /s, which are about 31 percent higher than ␣ 33 . As was suggested by the results of Doss and Wright ͓16͔ for PVC, this is likely the result of extrusion processes during the manufacturing of the neoprene sheets. Indeed, Choy et al. ͓27͔ found an increased diffusivity in the draw direction and decreased diffusivity in the direction normal to the draw of highly drawn polyethylene using a onedimensional flash method. At the larger deformation (ϭ1.52), the mean value of ␣ 33 is found to be 0.112 mm 2 /s, about 5.9 percent lower than the undeformed value. In contrast, the in-plane values increase to 0.165 and 0.159 mm 2 /s. It should be noted that the specimen appeared to be partially translucent to the flash energy at the stretched state. This leads to error in the indicated value of ␣ 33 because the boundary value problem used to determine ␣ 33 from the temperature history is no longer strickly valid. If such error is small, it reveals itself by the indicated value of ␣ 33 changing as the fraction of the temperature history used in data reduction increases. Taylor ͓24͔ showed that in such circumstances the value of ␣ 33 may be found by extrapolating the indicated values of ␣ 33 calculated using different fractions of the temperature history to the one indicated at zero fraction of the temperature history. This method of correction was employed at the high stretch states. The values of ␣ 11 and ␣ 22 remained independent of the fraction of the temperature history used in data reduction because they are strongly dependent on the temperature histories of the thermocouples that are outside the projection of the aperture opening. Conclusions The ability to measure the multiaxial mechanical response of elastomers, and similarly planar soft tissues, promises more complete data for the formulation of constitutive models for finite strain thermomechanics. The active control of the finite deformation, via the real-time video feedback to the motor controllers, allows measurement of the material response to a wide variety of deformations that theory reveals would be useful. The equibiaxial, proportional, and constant stretch tests discussed here have illustrated this capability. Reprogramming the motor control algorithm would allow constant invariant ͑cf. ͓12͔͒ and other tests to be easily performed, as well. Data from such multiaxial tests are not readily available in the literature. Formulating models of general thermoelastic response requires that the temperature field within the material also be modeled. To this end, this device incorporates an extension of the flash thermal diffusivity technique that allows measurement of the in-plane components of diffusivity as well as the more commonly measured out-of-plane component. Data presented illustrate that neo-