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

“…As an example, a third order truncation necessitates the determination of 9 material parameters. For some authors, the so-called Rivlin representation of W can be improved by considering other strain invariants 15,27 . Nevertheless, this form of strain energy is classically used for very large strain problems.…”

Section: Phenomenological Modelsmentioning

confidence: 99%

Comparison of Hyperelastic Models for Rubber-Like Materials

Marckmann

Verron

2006

Rubber Chemistry and Technology

554

307

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The present paper proposes a thorough comparison of twenty hyperelastic models for rubber-like materials. The ability of these models to reproduce different types of loading conditions is analyzed thanks to two classical sets of experimental data. Both material parameters and the stretch range of validity of each model are determined by an efficient fitting procedure. Then, a ranking of these twenty models is established, highlighting new efficient constitutive equations that could advantageously replace well-known models, which are widely used by engineers for finite element simulation of rubber parts.

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“…[6 -8] are essential for the proofs that follow and are all found in the work of Criscione et al (6).…”

Section: Appendix Bmentioning

confidence: 99%

“…No two scalar measures (e.g., tensor norm and FA) can account for the degrees of freedom represented by the three eigenvalues. In order to describe the third degree of freedom we adopt the term "tensor mode" from Criscione et al (6) to refer to a shape metric that quantifies the type of tensor anisotropy, as it ranges from linear anisotropy, to orthotropy, to planar anisotropy. Tensor mode has not been described in the DT-MRI literature even though it orthogonally complements common tensor magnitude and anisotropy measures.…”

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confidence: 99%

Orthogonal tensor invariants and the analysis of diffusion tensor magnetic resonance images

Ennis

Kindlmann

2005

Magnetic Resonance in Med

248

230

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This paper outlines the mathematical development and application of two analytically orthogonal tensor invariants sets. Diffusion tensors can be mathematically decomposed into shape and orientation information, determined by the eigenvalues and eigenvectors, respectively. The developments herein orthogonally decompose the tensor shape using a set of three orthogonal invariants that characterize the magnitude of isotropy, the magnitude of anisotropy, and the mode of anisotropy. The mode of anisotropy is useful for resolving whether a region of anisotropy is linear anisotropic, orthotropic, or planar anisotropic. Both tensor trace and fractional anisotropy are members of an orthogonal invariant set, but they do not belong to the same set. It is proven that tensor trace and fractional anisotropy are not mutually orthogonal measures of the diffusive process Key words: DT-MRI; tractography; tensor; invariant; anisotropyThe utility of magnetic resonance imaging (MRI) to characterize biologic tissue is amplified by analysis and visualization methods that help researchers to see and understand the structures within the data. Tensor-valued imaging is an increasingly important source of information about tissue structure and dynamics, such as with diffusion tensor MRI (DT-MRI), and strain tensors derived from displacement encoded MRI. An effective and established way of describing structure in tensor fields is through a function that maps tensors to more readily understood scalar metrics. The medical imaging literature provides a variety of such metrics (e.g., bulk mean diffusivity, fractional anisotropy).This paper uses a combination of mathematics and visualizations to generate a rigorous and intuitive exposition of tensor shape, characterized by sets of orthogonal tensor invariants. Each invariant set decomposes tensor shape with an orthogonal basis. When invariants have application-specific significance (as they do in DT-MRI imaging), orthogonality is a useful property of an invariant set, because it isolates the measurement of variation in one physiologic property from variations in another. These sets of orthogonal tensor invariants are then used to create informative visualizations of tensor field structure. The result is the establishment of two sets of orthogonal tensor invariants that incorporate the established use of fractional anisotropy (FA) and apparent diffusion coefficient (ADC).Diffusion tensors are symmetric and thus can be decomposed into an eigensystem of three real eigenvalues and three mutually orthogonal eigenvectors. We adopt the terminology that tensor "shape" refers to those degrees of freedom in tensor values (components of the matrix representation of a tensor) spanned by changes in the eigenvalues, while keeping eigenvectors fixed. "Orientation," on the other hand, refers to the complementary degrees of freedom associated with changes in the eigenvectors, while keeping the eigenvalues fixed. Defining shape and orientation in terms of the tensor eigensystem coincides with the standard visuali...

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

Cited by 177 publications

References 16 publications

The effects of noise over the complete space of diffusion tensor shape

The effects of noise over the complete space of diffusion tensor shape

Comparison of Hyperelastic Models for Rubber-Like Materials

Orthogonal tensor invariants and the analysis of diffusion tensor magnetic resonance images

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