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Table 4 Layer-specific circumferential and axial components of the deformation gradients \({\mathbf{G}}^{k}\), \({\mathbf{F}}_{1}\) and \({\mathbf{F}}_{{\mathrm{residual}}, k}={\mathbf{F}}_{1}{\mathbf{G}}^{k}.\)

From: The Role of Layer-Specific Residual Stresses in Arterial Mechanics: Analysis via a Novel Modelling Framework

  Intima Media Adventitia
\({\mathbf{G}}^{k}\) \({\widehat{\lambda }}_{{\mathbb{X}}}^{{\mathrm{i}}}\) \({\widehat{\lambda }}_{{\mathbb{Z}}}^{{\mathrm{i}}}\) \({\widehat{\lambda }}_{{\mathbb{X}}}^{{\mathrm{m}}}\) \({\widehat{\lambda }}_{{\mathbb{Z}}}^{{\mathrm{m}}}\) \({\widehat{\lambda }}_{{\mathbb{X}}}^{{\mathrm{a}}}\) \({\widehat{\lambda }}_{{\mathbb{Z}}}^{{\mathrm{a}}}\)
I 1.04 1.02 1.04 0.99 0.94 1.02
II 1.02 1.05 1.03 0.97 0.98 0.99
III 1.03 1.00 1.01 0.99 0.97 1.02
Mean \(\pm\) SD 1.03 \(\pm\) 0.01 1.03 \(\pm\) 0.02 1.03 \(\pm\) 0.01 0.98 \(\pm\) 0.01 0.96 \(\pm\) 0.01 1.01 \(\pm\) 0.01
\({\mathbf{F}}_{1}\) \({\Lambda }_{\Theta }^{{\mathrm{i}}}\) \({\Lambda }_{Z}^{{\mathrm{i}}}\) \({\Lambda }_{\Theta }^{{\mathrm{m}}}\) \({\Lambda }_{Z}^{{\mathrm{m}}}\) \({\Lambda }_{\Theta }^{{\mathrm{a}}}\) \({\Lambda }_{Z}^{{\mathrm{a}}}\)
I 0.89 1.00 0.98 1.00 1.09 1.00
II 0.92 1.00 0.98 1.00 1.07 1.00
III 0.92 1.00 0.98 1.00 1.07 1.00
Mean \(\pm\) SD 0.91 \(\pm\) 0.01 1.00 \(\pm\) 0.00 0.98 \(\pm\) 0.00 1.00 \(\pm\) 0.00 1.07 \(\pm\) 0.01 1.00 \(\pm\) 0.00
\({\mathbf{F}}_{{\mathrm{residual}}, k}={\mathbf{F}}_{1}{\mathbf{G}}^{k}\) \({\widehat{\lambda }}_{{\mathbb{X}}}^{{\mathrm{i}}}{\Lambda }_{\Theta }^{{\mathrm{i}}}\) \({{\widehat{\lambda }}_{{\mathbb{Z}}}^{{\mathrm{i}}}\Lambda }_{Z}^{{\mathrm{i}}}\) \({\widehat{\lambda }}_{{\mathbb{X}}}^{{\mathrm{m}}}{\Lambda }_{\Theta }^{{\mathrm{m}}}\) \({{\widehat{\lambda }}_{{\mathbb{Z}}}^{{\mathrm{m}}}\Lambda }_{Z}^{{\mathrm{m}}}\) \({{\widehat{\lambda }}_{{\mathbb{X}}}^{{\mathrm{a}}}\Lambda }_{\Theta }^{{\mathrm{a}}}\) \({{\widehat{\lambda }}_{{\mathbb{Z}}}^{{\mathrm{a}}}\Lambda }_{Z}^{{\mathrm{a}}}\)
I 0.92 1.02 1.02 0.99 1.03 1.02
II 0.93 1.06 1.01 0.98 1.04 1.00
III 0.95 1.00 1.00 0.99 1.03 1.02
Mean \(\pm\) SD 0.93 \(\pm\) 0.01 1.03 \(\pm\) 0.02 1.01 \(\pm\) 0.01 0.98 \(\pm\) 0.00 1.04 \(\pm\) 0.00 1.01 \(\pm\) 0.01
  1. \({\mathbf{F}}_{{\mathrm{residual}}, k}\) defines the residual stretches experienced by each layer in the unload cylindrical vessel configuration (\({\kappa }_{\mathrm{unloaded}}\))