Nonuniform Stress Field Determination Based on Deformation Measurement

We demonstrate a technique that, under certain circumstances, will determine stresses associated with a nonuniform deformation field without knowing the detailed constitutive behavior of the deforming material. This technique is based on (1) a detailed deformation measurement of a domain and (2) the observation that for isotropic materials, the strain and the stress, which form the so-called work-conjugate pair, are co-axial, or their eigenvectors share the same direction. The particular measures for strain and stress considered are the Lagrangian strain and the second Piola-Kirchhoff stress. The deformation measurement provides the field of the principal stretch orientation θλ and since the Lagrangian strain and the second Piola-Kirchhoff stress are co-axial, the principal stress orientation θs of the second Piola-Kirchhoff stress is determined. The Cauchy stress is related to the second Piola-Kirchhoff stress through the deformation gradient tensor, which can be measured experimentally. We then show that the principal stress orientation θσ of the Cauchy stress is the sum of the principal stretch orientation θλ and the local rigid-body rotation θq, which is determinable by the deformation gradient through polar decomposition. With the principal stress orientation θσ known, the equation of equilibrium, now in terms of the two principal stresses σ1 and σ2, and θσ, can be solved numerically with appropriate traction boundary conditions. The technique is then applied to the experimental case of nonuniform deformation of a PVC sheet with a circular hole and subject to tension. Limitations and restrictions of the technique and possible extensions will be discussed.

Design Arbitrary Shaped 2D Acoustic Cloak Without Singularity

A method is proposed to design arbitrary shaped two dimensional (2D) isotropic-inertia acoustic cloaks without singularity. The method is based on the deformation view of the transformation method, where the transformation tensor A is identified as the deformation gradient tensor and the transformed material parameters can be expressed by the principal stretches in the principal system of the deformation. The infinite material parameters of a perfect 2D cloak is induced by an infinite principal stretch in one direction while the other two remains finite at the inner boundary during the transformation. To circumvent this difficulty, for a 2D cloak we can choose the principal stretch perpendicular to the cloak plane to be also infinite but in the same order as the infinite principal stretch in the cloak plane during the transformation, so the transformed material parameters may keep finite. To illustrate this idea, the analytical expressions of nonsingular material parameters for a cylindrical acoustic cloak are given. For the acoustic cloaks with irregular shapes, the numerical method is proposed to evaluate the principal stretches and in turn the nonsingular material parameters. The designed 2D cloaks are validated by numerical simulation.

Improved Prediction of the Collagen Fiber Architecture in the Aortic Heart Valve

Living tissues show an adaptive response to mechanical loading by changing their internal structure and morphology. Understanding this response is essential for successful tissue engineering of load-bearing structures, such as the aortic valve. In this study, mechanically induced remodeling of the collagen architecture in the aortic valve was investigated. It was hypothesized that, in uniaxially loaded regions, the fibers aligned with the tensile principal stretch direction. For biaxial loading conditions, on the other hand, it was assumed that the collagen fibers aligned with directions situated between the principal stretch directions. This hypothesis has already been applied successfully to study collagen remodeling in arteries. The predicted fiber architecture represented a branching network and resembled the macroscopically visible collagen bundles in the native leaflet. In addition, the complex biaxial mechanical behavior of the native valve could be simulated qualitatively with the predicted fiber directions. The results of the present model might be used to gain further insight into the response of tissue engineered constructs during mechanical conditioning.

Strain-rate formulation of ice fabric evolution

Reorientation of individual crystal-glide planes as isotropic surface ice is deformed during its passage to depth in an ice sheet, lattice rotation, creates a fabric and associated anisotropy. A simple macroscopic description is that these material glide planes are rotated towards planes normal to an axis of compression, and away from planes normal to an axis of extension, inducing an instantaneous orthotropic viscous response with reflexional symmetries in the planes orthogonal to the current principal stretch axes. An orthotropic viscous law is presented for the strain rate expressed in terms of the deviatoric stress, the deformation, and three structure tensors based on the principal stretch axes. This anisotropic relation is expressed in terms of a single fabric response function in addition to the isotropic ice viscosity. The predicted responses in uniaxial compression and simple shear are determined. While the uniaxial response yields an explicit relation between the axial strain rate and stress, it is found that the shear response is governed by three, complicated, coupled relations between the shear strain rate and three deviatoricstress components. The new result derived here is the solution of this system: an explicit relation between the shear strain rate and shear stress. Correlation of these relations with idealized uniaxial and shear responses is then used to determine the required fabric function in the model law.

Experimental Investigations and Numerical Modeling of Incompressible Elastomers during Non-Homogeneous Deformations

Three hyperelastic constitutive models, the Arruda and Boyce model, the Yeoh model, and the Ogden model, are used to simulate the response of vinyl elastomers in large, non-homogeneous deformation states. A single uniaxial compression test is used to characterize the elastomers to obtain model constants. Drucker stability is enforced by restriction of the coefficients for the Yeoh and Ogden models to produce physically feasible uniaxial results. The ability of the models to predict the global force versus deformation responses and deformed shapes for large deformation shear, tensile deformation of a long bar with fixed grips, and inflation of a thin disk is examined. The experiments used in this study are designed to provide known boundary conditions to eliminate ambiguity in the modeling. We demonstrate for the first time that each of these models may be sufficiently characterized via a simple, homogeneous compression test to allow accurate predictions of large, non-homogeneous deformations involving rotations of the principal stretch directions.

Large Plane-to-Surface Deformations of Membranes With Inclusion

For large deformations, the strain-energy density function for a neo-Hookean membrane is dominated by the sum of squares of the two principal stretch ratios. This property reduces the displacement equations of equilibrium for the class of problems considered to three uncoupled linear equations. The nonlinear coupling appears only in the algebraic stress calculations. In light of the scarcity of exact solutions to nontrivial problems, the approximate but explicit solutions obtained here should be of some practical value.