Figure 1
Global and corotational local systems of reference for the bar.
Figure 2
External and internal virtual work performed forces and due to a virtual displacement .
Figure 3
Unphysical behavior for large compressive strains (for stretch approaching zero) in the cases of and .
Figure 4
Cauchy stress versus stretch for different values of the Poisson ratio : (a) and (b) .
Figure 5
Cauchy stress versus stretch for incompressible hyperelastic and linear stress-strain relationships (.
Figure 6
A constant stress suddenly applied at instant , the resulting strain profile in time and the corresponding creep-compliance function.
Figure 7
Compliance is independent of the creep stress level.
Figure 8
Compliance for finite strains considering the linear relationship between (a) Green-Lagrange strain Second Piola-Kirchhoff stress and (b) engineering strain and stress, both with .
Figure 9
Simple and generalized Kelvin-Voigt rheological models.
Figure 10
Purely viscous system.
Figure 11
Unphysical behavior for large compressive strains (for stretch approaching zero) in the cases of and considering and
Figure 12
: Cauchy stress versus stretch for different values of the Poisson ratio in the viscoelastic model: (a) and (b)
Figure 13
(a) Two-dimensional truss structure: initial geometry and (b) load history.
Figure 14
2D truss: displacement history of node 2 for both bars viscoelastic. (a) Whole time range evaluated (from 0 to 1 seconds), (b) detail in the neighborhood of the snap-throughs and (c) the moment when curves change position.
Figure 15
2D truss: (a) Cauchy stress history for both bars viscoelastic; (b) detail of the plot between 0.15s and 0.3s.
Figure 16
2D truss: (a) Total logarithmic strain rate and (b) viscous logarithmic strain rate histories of node 2 for both bars viscoelastic.
Figure 17
2D truss: Displacement history of node 2 for both elastic bars with (a) Cauchy-Log, (b) Eng-Eng and (c) 2PK-GL material models.
Figure 18
2D truss: Cauchy stress history for both elements with (a) Cauchy-Log, (b) Eng-Eng and (c) 2PK-GL material models.
Figure 19
2D truss:(a) Displacement and (b) stress history of node 2 for both bars hyperelastic (undamped and damped).
Figure 20
2D truss structure with hyperelastic and viscoelastic materials: (a) horizontal and (b) vertical displacements of node 2.
Figure 21
2D truss structure with hyperelastic and viscoelastic materials: relationship between X and Y displacements
Figure 22
2D truss: Cauchy stress history (a) Viscoelastic element (b) Hyperelastic element.
Figure 14
2D truss: displacement history of node 2 for both bars viscoelastic. (a) Whole time range evaluated (from 0 to 1 seconds), (b) detail in the neighborhood of the snap-throughs and (c) the moment when curves change position.
Figure 15
2D truss: (a) Cauchy stress history for both bars viscoelastic; (b) detail of the plot between 0.15s and 0.3s.
Figure 16
2D truss: (a) Total logarithmic strain rate and (b) viscous logarithmic strain rate histories of node 2 for both bars viscoelastic.
Figure 17
2D truss: Displacement history of node 2 for both elastic bars with (a) Cauchy-Log, (b) Eng-Eng and (c) 2PK-GL material models.
Figure 18
2D truss: Cauchy stress history for both elements with (a) Cauchy-Log, (b) Eng-Eng and (c) 2PK-GL material models.
Figure 19
2D truss:(a) Displacement and (b) stress history of node 2 for both bars hyperelastic (undamped and damped).
Figure 20
2D truss structure with hyperelastic and viscoelastic materials: (a) horizontal and (b) vertical displacements of node 2.
Figure 21
2D truss structure with hyperelastic and viscoelastic materials: relationship between X and Y displacements
Figure 22
2D truss: Cauchy stress history (a) Viscoelastic element (b) Hyperelastic element.
Figure 23
Three-dimensional star truss structure: initial geometry and corresponding (a) nodes and elements, (b) front and perspective views and (c) load history.
Figure 24
3D truss: (a) vertical displacements of node 7 for all bars viscoelastic; (b) detail of the plot in the oscillatory region.
Figure 25
3D truss: some configurations for the viscoelastic case (Cauchy-Log material model).
Figure 26
3D truss: (a) Cauchy stress history for element 1. All bars viscoelastic; (b) detail of the plot in the oscillatory region.
Figure 27
3D truss: (a) Cauchy stress history for element 10. All bars viscoelastic; (b) detail of the plot in the oscillatory region.
Figure 28
3D truss: (a) Cauchy stress history for element 23. All bars viscoelastic; (b) detail of the plot in the oscillatory region.
Figure 32
3D truss: (a) vertical displacements of node 7 for all bars hyperelastic; (b) detail of the plot in the first 0.3 seconds.
Figure 33
3D truss: Stress history for element (a) 1, (b) 10 and (c) 23 with and without structural damping. All bars hyperelastic.
Figure 34
3D truss: Relationship between the displacements of node 7 in the (a) X and Z, (b) Y and Z and (c) X and Y directions. Mixed viscoelastic and hyperelastic bars.
Figure 35
Mixed 3D truss: Cauchy stress for viscoelastic element 4; (b) detail of the plot in the first 0.1 seconds.
Figure 36
Mixed 3D truss: (a) Cauchy stress for hyperelastic element 1; (b) detail of the plot in the first 0.1 seconds.
Figure 37
3D truss structure with hyperelastic and viscoelastic materials: behavior over time.
Figure 24
3D truss: (a) vertical displacements of node 7 for all bars viscoelastic; (b) detail of the plot in the oscillatory region.
Figure 25
3D truss: some configurations for the viscoelastic case (Cauchy-Log material model).
Figure 26
3D truss: (a) Cauchy stress history for element 1. All bars viscoelastic; (b) detail of the plot in the oscillatory region.
Figure 27
3D truss: (a) Cauchy stress history for element 10. All bars viscoelastic; (b) detail of the plot in the oscillatory region.
Figure 28
3D truss: (a) Cauchy stress history for element 23. All bars viscoelastic; (b) detail of the plot in the oscillatory region.
Figure 32
3D truss: (a) vertical displacements of node 7 for all bars hyperelastic; (b) detail of the plot in the first 0.3 seconds.
Figure 33
3D truss: Stress history for element (a) 1, (b) 10 and (c) 23 with and without structural damping. All bars hyperelastic.
Figure 34
3D truss: Relationship between the displacements of node 7 in the (a) X and Z, (b) Y and Z and (c) X and Y directions. Mixed viscoelastic and hyperelastic bars.
Figure 35
Mixed 3D truss: Cauchy stress for viscoelastic element 4; (b) detail of the plot in the first 0.1 seconds.
Figure 36
Mixed 3D truss: (a) Cauchy stress for hyperelastic element 1; (b) detail of the plot in the first 0.1 seconds.
Figure 37
3D truss structure with hyperelastic and viscoelastic materials: behavior over time.
Figure 29
3D truss: (a) total and (b) viscous logarithmic strain rates for Cauchy-Log material model.
Figure 30
3D truss: (a) total and (b) viscous logarithmic strain rates for Eng-Eng material model.
Figure 31
3D truss: (a) total and (b) viscous logarithmic strain rate for 2PK-GL material model.
Table 1
Tangent moduli for linear relationships between different stress-strain measures and internal virtual work evaluated using the conjugate pair .
Table 2
Viscoelastic material properties (Abdelrahman and El-Shafei, 2021Abdelrahman, A.A., El-Shafei, A.G. (2021). Modeling and analysis of the transient response of viscoelastic solids, Waves in Random and Complex Media 31: 1990—2020.)
Table 3
Hyperelastic material properties of natural rubber (Endo et al., 2021Endo, V. T., Fancello, E. A., Muñoz-Rojas, P. A. (2021). A study on the computational effort of hyper-dual numbers to evaluate derivatives in geometrically nonlinear hyperelastic trusses, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 43: 1-15.)