On the use of convected coordinate systems in the mechanics of continuous media derived from a QR factorization of F

Freed, Alan D.; Zamani, Shahla

doi:10.1016/j.ijengsci.2018.02.011

Cited by 16 publications

(4 citation statements)

References 19 publications

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“…Volume change is measured by J = det T = det Λ = abc. Parameters a, b, c, α, β, γ are physical attributes because they can be measured directly by an experimentalist in the physical coordinate system of later (2.12), without post analysis [38,40,41]. Let C = F T F = T T T = C IJ e I ⊗ e J = F i I δ ij F j J e I ⊗ e J = T α I δ αβ T β J e I ⊗ e J (2.9) denote the symmetric right Cauchy-Green deformation tensor.…”

Section: Kinematics Of Finite Deformationmentioning

confidence: 99%

A constitutive model for lung mechanics and injury applicable to static, dynamic, and shock loading

Clayton

Freed

2020

Mech Soft Mater

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A continuum model for lung parenchyma is constructed. The model describes the thermomechanical response over a range of loading rates-from static to dynamic to shock waves-and a range of stress states, including isotropic expansion, triaxial extension, simple shear, and plane wave compression. Nonlinear elasticity, viscoelasticity, and damage are included, with the latter associated with changes of biological function as well as mechanical stiffness. A Gram-Schmidt decomposition of the deformation gradient leads to strain attributes that enter the thermodynamic potentials as state variables. A free energy function is designed for loading at low to moderate rates and tensile pressures, whereby the tissue response, with surface tension, is preeminent. An internal energy function is designed for wave propagation analysis, including shock waves, whereby compressibility of the air inside the alveoli is addressed via a composite stiffness based on a closed-cell assumption. The model accurately represents the response to triaxial loading, pressure relaxation, and dynamic torsion with relatively few parameters. Longitudinal wave speeds are reasonable for ranges of internal airway pressure and transpulmonary pressure. Airway pressure strongly affects the response to plane wave compression. Criteria for local injury and damage progression depend on a normalized energy density and its gradient, where the latter is paramount for impact problems involving fast pressure rises. Results suggest that local damage associated with edema is induced at load intensities much lower than those that would cause significant stiffness changes due to rupture, major tearing, or local collapse.

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Section: Kinematics Of Finite Deformationmentioning

confidence: 99%

A constitutive model for lung mechanics and injury applicable to static, dynamic, and shock loading

Clayton

Freed

2020

Mech Soft Mater

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“…Laplace stretch, as it has been used in the literature to date, e.g., [1,2,3,4,12,5,6,13,8,14,11,7], derives from a Gram-Schmidt (or QR) decomposition of the deformation gradient F, where matrix Q is orthogonal, and matrix R is upper triangular. Given a coordinate system with base vectors ( e 1 , e 2 , e 3 ), we denote such a decomposition as F = RU , where R = R ij e i ⊗ e j has orthogonal components, and U = U ij e i ⊗ e j has uppertriangular components.…”

Section: Laplace Stretchmentioning

confidence: 99%

“…whose constituents are measured in a coordinate frame with base vectors [13] e all of which are described in terms of physical attributes defined as…”

Section: Lagrangian Stretch Attributesmentioning

confidence: 99%

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Laplace Stretch: Eulerian and Lagrangian Formulations

Freed¹,

Zamani²,

Szabó³

et al. 2020

Preprint

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Two triangular factorizations of the deformation gradient tensor are studied. The first, termed the Lagrangian formulation, consists of an upper-triangular stretch premultiplied by a rotation tensor. The second, termed the Eulerian formulation, consists of a lower-triangular stretch postmultiplied by a different rotation tensor. The corresponding stretch tensors are denoted as the Lagrangian and Eulerian Laplace stretches, respectively. Kinematics (with physical interpretations) and work conjugate stress measures are analyzed and compared for each formulation. While the Lagrangian formulation has been used in prior work for constitutive modeling of anisotropic and hyperelastic materials, the Eulerian formulation, which may be advantageous for modeling isotropic solids and fluids with no physically identifiable reference configuration, does not seem to have been used elsewhere in a continuum mechanical setting.

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