Nonlinear dynamics of loaded visco-hyperelastic spherical shells

Abstract

In this paper, the nonlinear dynamic behaviors, especially, limit cycles and chaos, are investigated for the spherical shell composed of a class of visco-hyperelastic materials subjected to uniform radial loads at its inner and outer surfaces. To include the thickness effect, a more general model compared with the membrane and thin plate is proposed to investigate the dynamic characteristics of the visco-hyperelastic structure. Then, the coupled integro-differential equations describing the radially symmetric motion of the spherical shell are derived in terms of the variational principle and the finite viscoelasticity theory. Due to both the geometrical and physical nonlinearities, there exists an asymmetric homoclinic orbit for the hyperelastic structure. Particularly, under constant loads, the system converges to a stable equilibrium point, and the convergence position and speed are closely related to both the initial condition and the viscosity because of the existence of different basins, while under periodic loads, some complex phenomena, such as the limit cycles and chaos, are found, and the chaotic phenomena are analyzed by the bifurcation diagram and Lyapunov exponent. Moreover, by numerical analyses, parametric studies are carried out to illustrate the effects of viscosity, load amplitude, external frequency and initial condition.

Appendix

In the viscoelastic model (Fig. 1), the strain energy functions of the two hyperelastic springs are

$$ W_{\alpha } = \hat{W}\left( {{\mathbf{B}}_{\alpha } } \right) = - \frac{{\mu^{\alpha } J_{m}^{\alpha } }}{2}\log \left( {1 - \frac{{I_{1}^{\alpha } - 3}}{{J_{m}^{\alpha } }}} \right),\;W_{\beta } = \hat{W}\left( {{\mathbf{B}}_{\beta } } \right) = - \frac{{\mu^{\beta } J_{m}^{\beta } }}{2}\log \left( {1 - \frac{{I_{1}^{\beta } - 3}}{{J_{m}^{\beta } }}} \right) $$

(A1)

where ${\mathbf{B}}_{\alpha } = {\mathbf{F}}_{\alpha } \cdot {\mathbf{F}}_{\alpha }^{T}$ and ${\mathbf{B}}_{\beta } = {\mathbf{F}}_{\beta } \cdot {\mathbf{F}}_{\beta }^{T}$ are the left Cauchy–Green deformation tensors, $I_{1}^{\alpha }$ and $I_{1}^{\beta }$ are the first invariants. ${\varvec{\upsigma}}_{\alpha }$, ${\varvec{\upsigma}}_{\beta }$ and ${\varvec{\upsigma}}_{\gamma }$ are Cauchy stress tensors in the viscoelastic model, and the following relation is satisfied for the Maxwell model

$$ {\varvec{\upsigma}}_{\beta } = {\varvec{\upsigma}}_{\gamma } $$

(A2)

Additionally, the total stress is given by

$$ {\varvec{\upsigma}} = {\varvec{\upsigma}}_{\alpha } + {\varvec{\upsigma}}_{\beta } $$

(A3)

and the multiplication decomposition of the deformation gradient tensor is that

$$ {\mathbf{F}} = {\mathbf{F}}_{\alpha } = {\mathbf{F}}_{\beta } \cdot {\mathbf{F}}_{\gamma } $$

(A4)

In the finite element implement, the following form of the strain energy function is considered to deal with the incompressibility

$$ W = \hat{W}\left( {I_{1} ,J} \right) = \hat{W}_{D} \left( {\bar{I}_{1} } \right) + \hat{W}_{V} \left( J \right) = \mu W_{0} \left( {\bar{I}_{1} } \right) + K\left( {\frac{{J^{2} - 1}}{2} - \ln J} \right) $$

(A5)

where $\mu$ and $K$ are material parameters, $J = \det \left( {\mathbf{F}} \right)$, $\bar{I}_{1} = J^{ - 2/3} I_{1}$ and

$$ W_{0} \left( {\bar{I}_{1} } \right) = - \frac{{J_{m} }}{2}\ln \left( {1 - \frac{{\bar{I}_{1} - 3}}{{J_{m} }}} \right) $$

(A6)

The Cauchy stress tensors of springs $\alpha$ and $\beta$ are

$$ {\varvec{\upsigma}}_{\alpha } = \frac{2}{J}\frac{{\partial W_{\alpha } }}{{\partial {\mathbf{B}}}} \cdot {\mathbf{B}},\;{\varvec{\upsigma}}_{\beta } = \frac{2}{J}\left[ {\frac{{\partial W_{\beta } }}{{\partial {\mathbf{B}}_{\beta } }} \cdot {\mathbf{B}}_{\beta } } \right]_{EQ} $$

(A7)

The material Jacobian could be evaluated as follows

$$ {\mathbf{c}} = {\mathbf{c}}_{\alpha } + {\mathbf{c}}_{\beta } = \frac{4}{J}\frac{\partial }{{\partial {\mathbf{B}}}}\left( {\frac{{\partial W_{\alpha } }}{{\partial {\mathbf{B}}}} \cdot {\mathbf{B}}} \right) \cdot {\mathbf{B}} + \frac{4}{J}\left[ {\frac{\partial }{{\partial {\mathbf{B}}_{\beta } }}\left( {\frac{{\partial W_{\beta } }}{{\partial {\mathbf{B}}_{\beta } }} \cdot {\mathbf{B}}_{\beta } } \right) \cdot {\mathbf{B}}_{\beta } } \right]_{EQ} $$

(A8)

where the subscript ‘EQ’ means that the stress of spring $\beta$ depends on the solution of the following evolution equation

$$ {\dot{\mathbf{B}}}_{\beta } = {\mathbf{LB}}_{\beta } + {\mathbf{B}}_{\beta } {\mathbf{L}}^{T} - {\mathbf{M}}:2{\varvec{\upsigma}}_{\beta } \cdot {\mathbf{B}}_{\beta } $$

(A9)

where

$$ {\mathbf{M}} = \frac{1}{{2\eta_{D} }}\left( {{\mathbf{1}}^{4} - \frac{1}{3}{\mathbf{1}} \otimes {\mathbf{1}}} \right) + \frac{1}{{9\eta_{V} }}{\mathbf{1}} \otimes {\mathbf{1}} $$

(A10)

is the mobility tensor. The main objective of the evolution equation is to determine the variable of the Maxwell model in order that the stress may be evaluated when the material motion is known. The main feature of the algorithm for the integration is to carry out an operator split of ${\dot{\mathbf{ B}}}_{\beta}$ into an elastic predictor $E$ and an inelastic corrector $I$, i.e.,

$$ {\dot{\mathbf{B}}}_{\beta } = \underbrace {{{\mathbf{LB}}_{\beta } + {\mathbf{B}}_{\beta } {\mathbf{L}}^{T} }}_{E} + \underbrace {{{\mathbf{F}}\mathop {{\mathbf{C}}_{\gamma }^{ - 1} }\limits^{.} {\mathbf{F}}^{T} }}_{I} $$

(A11)

In the elastic predictor step, let $\mathop {{\mathbf{C}}_{\gamma }^{ - 1} }\limits^{.} = 0$, then

$$ \left( {C_{i}^{ - 1} } \right)_{\text{trial}} = \left( {C_{i}^{ - 1} } \right)_{{t = t_{n - 1} }} $$

(A12)

$$ ({\mathbf{B}}_{\beta } )_{\text{trial}} = {\mathbf{F}}_{{t = t_{n} }} \cdot ({\mathbf{C}}_{\gamma }^{ - 1} )_{{t = t_{n - 1} }} \cdot {\mathbf{F}}_{{t = t_{n} }}^{T} $$

(A13)

In the inelastic corrector step, let ${\mathbf{L}} = 0$, then

$$ {\dot{\mathbf{B}}}_{\beta } = - \left( {2{\mathbf{M}}:{\varvec{\upsigma}}_{\beta } } \right) \cdot {\mathbf{B}}_{\beta } $$

(A14)

Solving the differential equation by the exponential mapping and taking the first-order accurate yield

$$ \varepsilon_{A\beta } = - \Delta t\left( {\frac{1}{{2\eta_{D} }}{\text{dev}}[{\varvec{\upsigma}}_{A} ] + \frac{1}{{9\eta_{V} }}{\varvec{\upsigma}}_{\beta } :{\mathbf{1}}} \right) + (\varepsilon_{A\beta } )_{\text{trial}} ,$$

(A15)

where ${\text{dev}}[{\varvec{\upsigma}}_{A} ]$ (A = 1, 2, 3) are the principal values of the deviatoric stress of ${\varvec{\upsigma}}_{\beta }$, $\varepsilon_{A\beta } = \ln \lambda_{A\beta }$ are the elastic logarithmic principal stretches. Additionally, Eq. (A15) can be used to determine the principal values of ${\mathbf{B}}_{\beta }$, and it can be solved by a local Newton iteration.