Andrew S. Douglas

An invariant basis for natural strain which yields orthogonal stress response terms in isotropic hyperelasticity

A novel constitutive formulation is developed for finitely deforming hyperelastic materials that exhibit isotropic behavior with respect to a reference configuration. The strain energy per unit reference volume, W, is defined in terms of three natural strain invariants, K1–3, which respectively specify the amount-of-dilatation, the magnitude-of-distortion, and the mode-of-distortion. Distortion is that part of the deformation that does not dilate. Moreover, pure dilatation (K2=0), pure shear (K3=0), uniaxial extension (K3=1), and uniaxial contraction (K3=−1) are tests which hold a strain invariant constant. Through an analysis of previously published data, it is shown for rubber that this new approach allows W to be easily determined with improved accuracy. Albeit useful for large and small strains, distinct advantage is shown for moderate strains (e.g. 2–25%). Central to this work is the orthogonal nature of the invariant basis. If η represents natural strain, then {K1,K2,K3} are such that the tensorial contraction of (∂Ki/∂η) with (∂Kj/∂η) vanishes when i≠j. This result, in turn, allows the Cauchy stress t to be expressed as the sum of three response terms that are mutually orthogonal. In particular (summation implied) t=Ai∂W/∂Ki, where the ∂W/∂Ki are scalar response functions and the Ai are kinematic tensors that are mutually orthogonal.

Description of the deformation of the left ventricle by a kinematic model.

A model of left ventricular (LV) kinematics is essential to identify the fundamental physiological modes of LV deformation during a complete cardiac cycle as observed from the motion of a finite number of markers embedded in the LV wall. Kinematics can be described by a number of modes of motion and deformation in succession. An obvious mode of LV deformation is the ejection of cavity volume while the wall thickens. In the more sophisticated model of LV kinematics developed here, seven time-dependent parameters were used to describe not only volume change but also torsion and shape changes throughout the cardiac cycle. Rigid-body motion required another six parameters. The kinematic model employed a deformation field that had no singularities within the myocardium, and all parameters describing the modes of deformation were dimensionless. Note that torsion, volume and symmetric shape changes all require the definition of a cardiac coordinate system, which has generally been related to the measured cardiac geometry by reference to approximate anatomical landmarks. However, in the present study the coordinate system was positioned objectively by a least-squares fit of the kinematic model to the measured motion of markers. Theoretically, at least five markers are needed to find a unique set of parameters.(ABSTRACT TRUNCATED AT 250 WORDS)

Physically based strain invariant set for materials exhibiting transversely isotropic behavior

Abstract A novel set of 5 strain invariants for materials exhibiting transversely isotropic behavior with respect to a reference configuration is developed through analysis of physical attributes of deformation. Experimental advantage for hyperelastic materials is demonstrated by showing that common tests can directly determine terms in W , the strain energy per unit reference volume. An analysis of symmetry allows the general form of W to be refined a priori. Moreover, this kinematics framework is potentially useful for solving inverse problems since the 5 response terms in the Cauchy stress t are mostly orthogonal (9 of the 10 mutual inner products vanish). For small deformation they are fully orthogonal (all 10 inner products vanish). A response term in t consists of an invariant response function multiplied by its associated kinematic tensor.

Modeling the motion of the internal tongue from tagged cine-MRI images

A new technique, tagged Cine-Magnetic Resonance Imaging (tMRI), was used to develop a mechanical model that represented local, homogeneous, internal tongue deformation during speech. The goal was to infer muscle activity within the tongue from tissue deformations seen on tMRI. Measurements were made in three sagittal slices (left, middle, right) during production of the syllable /ka/. Each slice was superimposed with a grid of tag lines, and the approximately 40 tag line intersections were tracked at 7 time-phases during the syllable. A local model, similar to a finite element analysis, represented planar stretch and shear between the consonant and vowel at 110 probed locations within the tongue. Principal strains were calculated at these locations and revealed internal compression and extension patterns from which inferences could be drawn about the activities of the Verticalis, Hyoglossus, and Superior Longitudinal muscles, among others.

Cardiac motion simulator for tagged MRI

Describes a computational simulator for use in cardiac imaging using tagged magnetic resonance imaging. The simulator incorporates a 13-parameter model of left-ventricular motion due to Arts et al. (1992) and applies it to a confocal prolate spherical shell, resembling the shape of the left ventricle. Using parameters determined in other work, our model can be made to assume a configuration representing one of 60 phases in the cardiac cycle. In this paper we determine the inverse motion map analytically, allowing pointwise correspondences to be made between two points at any two times. Using this mathematical relationship, we simulate the (tagged) magnetic resonance imaging process using a standard (tagged) spin-echo imaging equation. Image sequences can be synthesized at arbitrary orientations at any phase. We currently synthesize a SPAMM tag pattern with arbitrary spatial frequency, but other patterns can be readily incorporated. To accommodate two-dimensional motion estimation algorithms, we have created a two-dimensional simulator which restricts the three-dimensional motion to two dimensions. In either two or three dimensions, a true motion is output so that motion estimation algorithms can be compared against the truth. We conclude with a simple demonstration of the performance of the simulator.

Needle-tissue interaction forces for bevel-tip steerable needles

The asymmetry of a bevel-tip needle results in the needle naturally bending when it is inserted into soft tissue. As a first step toward modeling the mechanics of deflection of the needle, we determine the forces at the bevel tip. In order to find the forces acting at the needle tip, we measure rupture toughness and nonlinear material elasticity parameters of several soft tissue simulant gels and chicken tissue. We incorporate these physical parameters into a finite element model that includes both contact and cohesive zone models to simulate tissue cleavage. We investigate the sensitivity of the tip forces to tissue rupture toughness, linear and nonlinear tissue elasticity, and needle tip bevel angle. The model shows that the tip forces are sensitive to the rupture toughness. The results from these studies contribute to a mechanics-based model of bevel-tip needle steering, extending previous work on kinematic models.

The influence of inertia on elastic-plastic antiplane-shear crack growth

Abstract Results of a study of steady-state antiplane-shear crack growth in an elastic ideally-plastic material are reported. First, an exact expression for the distribution of strain on the crack line within the primary active plastic zone is obtained. It is shown that the expression reduces to the correct asymptotic form for the special case of vanishingly small distance from the crack tip for any nonzero crack growth speed, and it reduces to the correct limit as the crack speed approaches zero for any point on the crack line. Then, the full elastic-plastic deformation field is determined under the conditions of small-scale yielding by means of the numerical finite element method. The computed strain distribution on the crack line is found to compare closely with the analytical result for this distribution. Finally, the analytical and numerical results are combined with the “critical plastic strain at a characteristic distance” crack growth criterion to generate theoretical fracture toughness vs crack speed relationships. The results are similar to experimentally observed fracture toughness vs crack speed relationships. In particular, the results show a strong dependence of fracture toughness on crack speed for even moderate crack growth rates. Because the material response is independent of strain-rate, this suggests that the influence of inertia on the fracture resistance of the material is significant.

The mechanical properties of lead-titanate/polymer 0–3 composites

Abstract The stiffness and brittleness of pure piezoelectric ceramics limit the application of these materials as embedded sensors within composites. Thin-film compliant sensors have recently been developed by incorporating piezoelectric ceramic particles in a polymer matrix. When embedded in thicker composite structures, these thin-film sensors may be useful in monitoring internal mechanical conditions such as the evolution of damage. The mechanical properties of 0–3 composite films of calcium-modified lead titanate in Epon 828 epoxy and poly(vinylidene fluoride-trifluoroethylene) (P(VDF-TrFE)) matrices have been investigated in this work. The linear viscoelastic properties of these thin-film composites are measured, and the influences of strain, frequency, and ceramic content on these properties are investigated. Analytical and finite-element modeling are used to predict the effective viscoelastic properties of the composites, and these predictions are compared with the experimental results. The storage moduli predicted by these models are comparable to experimental results, while the predicted loss moduli are significantly less than the measured loss moduli. It is shown that the introduction of a third “interphase” layer at the ceramic/polymer boundary within the finite element model results in a better fit to the experimental data.

Macroscopic three-dimensional motion patterns of the left ventricle.

The pattern of displacements in the left ventricle (LV) can be described by 13 modes of motion and deformation. Three functional modes of deformation are essential for ejection: a decrease in cavity volume, torsion, and ellipticalization. Four additional modes are used to describe asymmetric deformation. Six modes of rigid body motion describe rotation and translation. In the LV 14-20 radiopaque markers were inserted in the wall of the LV. They were distributed more or less evenly from base to apex and around the circumference. Torsion and volume changes require the definition of a cardiac coordinate system. The point at which ejection focuses is used as the origin, and the torsion axis is used as the z-axis. In the present study the coordinate system was positioned objectively by a least squares fit of the kinematic model to the measured motion of markers. In five dogs in the control state the kinematic parameters were determined as a function of time for all 13 modes. The torsion axis was displaced 4 +/- 2 mm (mean +/- sd) from the center of the cross-section of the LV towards the lateral free wall. The direction of the torsion axis closely coincided with anatomical landmarks at the apex and base. During systole, a unique relation was found between the ratio of cavity volume to wall volume and torsion. This relation was universal to all LVs, the cylinder-symmetric mathematical model of cardiac mechanics inclusive. In diastole the patterns of deformation seem less universal and reproducible.

Mechanical characterization of active poly(vinyl alcohol)–poly(acrylic acid) gel

Abstract Active polymer gels expand and contract in response to certain environmental stimuli, such as the application of an electric field or a change in the pH level of the surroundings. This ability to achieve large, reversible deformations with no external mechanical loading has generated much interest in the use of these gels as actuators and “artificial muscles.” This work focuses on developing a means of characterizing the mechanical properties of active polymer gels and describing how these properties evolve as the gel actuates. Poly(vinyl alcohol)–poly(acrylic acid) (PVA–PAA) gel was chosen as the model material for this work because it is relatively simple and safe to both fabricate and actuate. PVA–PAA gels are fabricated on-site using a solvent-casting technique. These gels expand when moved from acidic to basic solutions, and contract when moved from basic to acidic solutions. The mechanical properties of the gel were characterized by conducting uniaxial tests on thin PVA–PAA gel films. A testing system has been developed which can measure stress and deformations of these films in a variety of liquid environments. The experimental results on PVA–PAA gels show these materials to be relatively compliant, and slightly viscoelastic and compressible. These gels are also capable of large recoverable deformations in both acidic and basic environments.