Elastic Kelvin-Poisson-Poynting solids described through scalar conjugate stress/strain pairs derived from a QR factorization of

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

“…(2.6) Greek indices α = 1, 2, 3 correspond to an intermediate configuration whose tangent vectors are pulled back from the spatial configuration by Q −1 or pushed forward from the reference configuration by T . The same global Cartesian basis is used for this configuration, called the "physical configuration" in [40]: Physically descriptive deformation measures are the three positive elongation ratios a, b, c that enter the extension matrix Λ and the three shear magnitudes α, β, γ that enter the simple shear matrix Γ . Volume change is measured by J = det T = det Λ = abc.…”

“…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.…”

“…Then, there is an ideal choice among coordinate bases for which the direction aligned with the rotated 1-axis and the direction normal to the rotated 1-2-plane remain invariant under a transformation by T [43]. A pivoting algorithm to achieve the corresponding upper triangular dominance of T is given in [40]. The physical basis of [40], denoted here by {g α }, corresponds to the following: g α = g = Q i α e i = Q i α δ J i e J , e i = (Q −1 ) α i g α = (Q T ) α i g α .…”

“…A pivoting algorithm to achieve the corresponding upper triangular dominance of T is given in [40]. The physical basis of [40], denoted here by {g α }, corresponds to the following: g α = g = Q i α e i = Q i α δ J i e J , e i = (Q −1 ) α i g α = (Q T ) α i g α . (2.12) The {g α } comprise a local Cartesian frame at each X.…”

“…. , 6 in terms of s αβ , are defined as follows [40]: Negative pressure per unit reference volume isτ 0 = −Jp, where Cauchy pressure is p = − 1 3 trσ . Normal stress differences areτ 1 ,τ 2 ,τ 3 .…”