Stability and Localization of Deformation Delay in Finitely Strained Plates at Arbitrary Strain-Rates

Abstract

Of interest here is the stability and associated deformation localization of structures when inertial effects are considered. Concerned primarily with the study of necking failure patterns, the prevailing approach in the relevant literature uses the “modal analysis” method to find the wavelength of the structure’s fastest growing eigenmode, an approach that often uses a rate-dependent material response. However, the experimental studies of (Zhang and Ravi-Chandar in Int. J. Fracture 142: 183–217, 2006; Zhang and Ravi-Chandar in Int. J. Fracture 163: 41–65, 2010) on the high strain-rate expansion of thin rings and tubes, show no evidence of a dominant wavelength in their failure mode and no influence of strain-rate sensitivity on the necking strains. Moreover, modal analysis assumes that at all times the entire structure sees the applied eigenmode perturbation in spite of the physical limitation of a finite wave propagation speed. In addition, the closely related problem of stability in dynamically loaded structures, i.e., the time evolution of perturbations introduced at different times during loading, does not seem to have attracted attention.

Based on the above-mentioned experimental and theoretical observations, (Ravi-Chandar and Triantafyllidis in Int. J. Solids Struct. 58: 301–308, 2015) proposed a “localized perturbation” approach to study the dynamic stability of an incompressible, nonlinearly elastic bar at different strain-rates by following the evolution of spatially localized small perturbations introduced at different times. The goal of the present work is to study the dynamic stability – linear and nonlinear – of rate-independent biaxially strained thin plates by following the evolution of spatially localized perturbations introduced at different times, to understand the initiation of the corresponding failure mechanisms. Our 2D linearized analysis of a thin plate under plane stress state, shows that these plates are stable until $\tau _{L}$, the dimensionless limit time corresponding to the loss of the uniformly strained plate’s stability. This result is supported by fully nonlinear calculations.

Our nonlinear numerical calculations also show an imperfection amplitude-dependent and biaxiality-dependent delay in the appearance of localization patterns in dynamically loaded plates for dimensionless times well beyond $\tau _{m}$, corresponding to the onset of loss of ellipticity in the constitutive law. Moreover, the failure patterns of these plates are studied numerically by following the time evolution of randomly distributed imperfections of different amplitudes.

Appendices

Appendix A: Constitutive Laws Adopted and Their Plane Stress Incremental Moduli

Although the expressions for the 2D plane stress incremental moduli are stated in [29], their derivation from 3D considerations is interesting and hence, for the sake of completeness of the presentation, the details of the corresponding derivations are given in this Appendix.

The analysis presented in Sect. 2 is general; any rate-independent constitutive law¹⁶ which can be put in the form of (2.5), can be accommodated. Results presented here correspond to the three such models: a hyperelastic deformation theory model of plasticity, the $J_{2}$ deformation theory model of [24] and a finite strain generalization of the $J_{2}$ flow theory. All models are fitted to the same power law uniaxial stress-strain curve and thus share the same principal solution. For further details on this construction, the interested reader is referred to [26]. For the small perturbations considered here, deviations from proportional loading are initially small and no unloading occurs in the perturbed plate until well after $\tau _{m}$ (the time corresponding to the loss of ellipticity onset in the unperturbed – uniform strain – solution), thus justifying their use for analyzing the plate’s stability.

Assuming material incompressibility, the [24] $J_{2}$ deformation and the $J_{2}$ flow theory models that can be put in the following 3D rate form

$$ \overset{\triangledown}{\boldsymbol{\sigma}} = \boldsymbol{C} : \boldsymbol{D} - \dot{p} \boldsymbol{I} , $$

(A.1)

where $\overset{\triangledown}{(\phantom{A})}$ denotes the Jaumann rate of the Cauchy stress tensor ($\overset{\triangledown}{\boldsymbol{\sigma}} = \dot{\boldsymbol{\sigma}} - \boldsymbol{\Omega} \cdot \boldsymbol{\sigma} + \boldsymbol{\sigma} \cdot \boldsymbol{\Omega}$ with $\boldsymbol{\Omega}$ the spin tensor), $\boldsymbol{D}$ is the strain rate tensor and $\dot{p}$ the hydrostatic pressure rate.

For the case of finite strains, the above 3D current configuration relation can be transformed into its reference configuration counterpart

$$ \dot{S}_{\mathit{ij}} = L_{\mathit{ijkl}} \dot{E}_{\mathit{kl}} - \dot{p} C_{\mathit{ij}}^{-1} ;\quad E_{\mathit{ij}} = \frac{1}{2} \big( C_{\mathit{ij}} - \delta _{\mathit{ij}} \big) , $$

(A.2)

where $\boldsymbol{S}$ is the second Piola-Kirchhoff stress, $\boldsymbol{E}$ its work-conjugate Green-Lagrange strain and the reference configuration components of the incremental moduli tensor $\boldsymbol{L}$ are

$$ \begin{aligned}[b] L_{\mathit{ijkl}} \equiv \,&\frac{2}{3} E^{*} \Big[\frac{1}{2}\big(C^{-1}_{\mathit{ik}} C^{-1}_{\mathit{jl}} + C^{-1}_{\mathit{il}} C^{-1}_{\mathit{jk}}\big) - \frac{3}{2} \Big(1 - \frac{E_{t}}{E^{*}}\Big) \frac{S^{\prime}_{\mathit{ij}}S^{\prime}_{\mathit{kl}}}{\sigma _{e}^{2}}\Big] \\ &{}- \frac{1}{2} \Big[C^{-1}_{\mathit{ik}} S_{\mathit{jl}} + C^{-1}_{\mathit{jk}} S_{\mathit{il}} + C^{-1}_{\mathit{il}} S_{\mathit{jk}} + C^{-1}_{\mathit{jl}} S_{\mathit{ik}} \Big] , \end{aligned} $$

(A.3)

where $\boldsymbol{S}^{\prime}$ is the deviatoric part of the stress tensor $\boldsymbol{S}$ and $\sigma _{e}$ the von Mises equivalent stress, namely

$$ S^{\prime}_{\mathit{ij}} = S_{\mathit{ij}} - \frac{1}{3} C^{-1}_{\mathit{ij}} C_{\mathit{kl}} S_{\mathit{kl}}, \quad \sigma _{e}^{2} = \frac{3}{2} C_{\mathit{ik}} C_{\mathit{jl}} S^{\prime}_{\mathit{ij}} S^{ \prime}_{\mathit{kl}} . $$

(A.4)

In the above expressions $E^{*} = E$ for the $J_{2}$ flow theory while for the $J_{2}$ deformation theory of [24] $E^{*} = E_{s} \equiv \sigma _{e} / \epsilon _{e}$ is the secant modulus of the uniaxial stress-strain curve. In both models $E_{t}$ is the tangent modulus of the uniaxial stress-strain curve $E_{t} \equiv \text{d}\sigma _{e} / \text{d} \epsilon _{e}$. For the $J_{2}$ hyperelastic model the general expression for $E^{*}$ is more complicated and will be given below for the special case of the biaxially stretched membrane.

The principal axes expressions in three dimensions for the (von Mises) equivalent stress $\sigma _{e}$ and the equivalent strain $\epsilon _{e}$, required for the calculation of $E_{s}$ and $E_{t}$ are

$$ \sigma _{e} = \big( \sigma _{1}^{2} + \sigma _{2}^{2} + \sigma _{3}^{2} - \sigma _{1} \sigma _{2} - \sigma _{2} \sigma _{3} - \sigma _{3} \sigma _{1} \big) ^{1/2} ,\quad \epsilon _{e} = \frac{2}{3} \big( \epsilon _{1}^{2} + \epsilon _{2}^{2} + \epsilon _{3}^{2} - \epsilon _{1} \epsilon _{2} - \epsilon _{2} \epsilon _{3} - \epsilon _{3} \epsilon _{1} \big) ^{1/2} . $$

(A.5)

Due to plane stress loading conditions

$$ \dot{S}_{3i} = 0 ,\quad \dot{E}_{\alpha 3} = 0 ,\quad \dot{E}_{33} = - C_{33} C^{-1}_{\gamma \delta} \dot{E}_{\gamma \delta} , $$

(A.6)

which substituted in (A.2) give the following relation between $\dot{S}_{\alpha \beta}$ and its work conjugate $\dot{E}_{\alpha \beta}$

$$ \dot{S}_{\alpha \beta} = {\mathcal{M}}_{\alpha \beta \gamma \delta} \dot{E}_{\gamma \delta} ; \quad {\mathcal{M}}_{\alpha \beta \gamma \delta} = L_{\alpha \beta \gamma \delta} - C_{33} \big( L_{\alpha \beta 33} C^{-1}_{ \gamma \delta} + C^{-1}_{\alpha \beta} L_{33 \gamma \delta} \big) + C^{-1}_{ \alpha \beta} C^{-1}_{\gamma \delta} C_{33}^{2} L_{3333} . $$

(A.7)

The above result, combined with (A.3) gives the following expression for $M_{\alpha \beta \gamma \delta}$

$$ \begin{aligned}[b] {\mathcal{M}}_{\alpha \beta \gamma \delta} = \,&\frac{2}{3} E^{*} \Big[ \frac{1}{2} \big(C^{-1}_{\alpha \gamma} C^{-1}_{\beta \delta} + C^{-1}_{ \alpha \delta} C^{-1}_{\beta \gamma} \big) + C^{-1}_{\alpha \beta} C^{-1}_{ \gamma \delta} - \frac{3}{2} \Big(1 - \frac{E_{t}}{E^{*}}\Big) \frac{S^{\prime}_{\alpha \beta}S^{\prime}_{\gamma \delta}}{\sigma _{e}^{2}} \Big] \\ &{}- \frac{1}{2} \Big[C^{-1}_{\alpha \gamma} S_{\beta \delta} + C^{-1}_{ \beta \gamma} S_{\alpha \delta} + C^{-1}_{\alpha \delta} S_{\beta \gamma} + C^{-1}_{\beta \delta} S_{\alpha \gamma} \Big] . \end{aligned} $$

(A.8)

Recalling the relations between $\boldsymbol{S}$ & $\boldsymbol{N}$ and $\boldsymbol{E}$ & $\boldsymbol{F}$, the plane stress incremental moduli $\boldsymbol{\mathcal{L}}$ in (2.5) are found to be

$$ \mathcal{L}_{\alpha \beta \gamma \delta} = {\mathcal {M}}_{\epsilon \beta \zeta \delta} F_{\alpha \epsilon} F_{\gamma \zeta} + S_{\delta \beta} \delta _{\alpha \gamma} . $$

(A.9)

For the case of biaxial loading of interest here $F_{\alpha \beta} = \operatorname{diag} \big( \lambda _{\gamma }\big)$ and thus

$$ C_{11} = \lambda _{1}^{2} , \quad C_{22} = \lambda _{2}^{2} , \quad C_{12} = 0 ; \quad S_{11} = \sigma _{1} / \lambda _{1}^{2} , \quad S_{22} = \sigma _{2} / \lambda _{2}^{2} , \quad S_{12} = 0 , $$

(A.10)

which upon substitution into (A.8), (A.9) gives the incremental moduli expressions in (A.16), (A.17) for $J_{2}$ flow and $J_{2}$ deformation theories.

Calculations of the moduli for the hyperelastic model use (A.9), (A.10) and $\boldsymbol{\mathcal{M}}$ is derived from the strain energy potential $W$

$$ {\mathcal {M}}_{\alpha \beta \gamma \delta} = \frac{\partial ^{2} W}{\partial E_{\alpha \beta} \partial E_{\gamma \delta}} = 4 \frac{\partial ^{2} W}{\partial C_{\alpha \beta} \partial C_{\gamma \delta}} , $$

(A.11)

obtained by successive application of chain rule of differentiation using $W(\epsilon _{e})$ where the equivalent strain $\epsilon _{e} = \epsilon _{e}(I_{1}, I_{2})$ is expressed in terms of the invariants of $\boldsymbol{C}$, which in turn depend on the principal stretch ratios $\lambda _{\alpha}$ through

$$ I_{1} = \operatorname{tr} \boldsymbol{C} = \lambda _{1}^{2} + \lambda _{2}^{2} ,\quad I_{2} = \det \boldsymbol{C} = \big( \lambda _{1} \lambda _{2} \big)^{2} . $$

(A.12)

The 3D strain energy density $W$ used here is a function of the equivalent strain $\epsilon _{e}$; the isotropic model is fitted to a piecewise power law uniaxial stress-strain curve¹⁷ resulting in the following expressions for $W$

$$ \frac{\sigma _{e}}{\sigma _{y}} = \Big( \frac{\epsilon _{e}}{\epsilon _{y}} \Big)^{m} \; ; \quad W = E\big( \epsilon _{y}\big)^{2} \bigg[ \frac{1}{1+m} \Big( \frac{\epsilon _{e}}{\epsilon _{y}} \Big)^{m+1} + \frac{1}{2} \Big( \frac{m-1}{m+1} \Big) \bigg] , \quad \left \{ \begin{aligned} & m= 1 \; \text{for} \; \epsilon _{e} \leq \epsilon _{y} , \\ & m = n \; \text{for} \; \epsilon _{e} > \epsilon _{y} , \end{aligned} \right . $$

(A.13)

where the equivalent strain $\epsilon _{e}$ is given in terms of the principal logarithmic strain components $\epsilon _{\alpha}$ by (A.5). Accounting for the incompressibility constraint $\epsilon _{1}+\epsilon _{2}+\epsilon _{3}=0$ (A.5) one obtains

$$ \sigma _{e} = \big(\sigma _{1}^{2} + \sigma _{2}^{2} - \sigma _{1} \sigma _{2} \big)^{1/2} \; , \quad \epsilon _{e} = \frac{2}{\sqrt{3}} \Big[ \epsilon _{1}^{2} + \epsilon _{2}^{2} + \epsilon _{1} \epsilon _{2} \Big] ^{1/2} \; , $$

(A.14)

where the equivalent stress $\sigma _{e}$ expression accounts for the plane stress condition $\sigma _{3}=0$ and the equivalent strain accounts for incompressibility $\epsilon _{1}+\epsilon _{2}+\epsilon _{3}=0$. Since the principal solution is biaxial straining, the principal Cauchy stresses $\sigma _{\alpha}$ are related to their work-conjugate the principal logarithmic strains $\epsilon _{\alpha}$ by:

$$ \sigma _{\alpha }= \frac{\partial W}{\partial \epsilon _{\alpha}} \; , \ \sigma _{1} = \frac{2}{3} E_{s} \big(2 \epsilon _{1} + \epsilon _{2} \big)\; , \ \sigma _{2} = \frac{2}{3} E_{s} \big( \epsilon _{1} + 2 \epsilon _{2} \big)\; ; \quad \epsilon _{\alpha}=\ln \lambda _{\alpha }\; , $$

(A.15)

where the secant $E_{s}$ and tangent $E_{t}$ moduli of the uniaxial stress-strain curve that appear in (A.3).

After some lengthy algebra we end in the following expressions for the non-zero components of the plane stress moduli in (2.6), given below in two groups; the normal moduli components are:

$$ \textstyle\begin{array}{rl} \mathcal{L}_{1111} & =\displaystyle \frac{1}{\lambda _{1}^{2}} \Big[ \frac{4}{3} E^{*} + \big( E_{t} - E^{*} \big) \Big( \frac{\sigma _{1}}{\sigma _{e}} \Big)^{2} - \sigma _{1} \Big] , \\ \mathcal{L}_{1122} = \mathcal{L}_{2211} & =\displaystyle \frac{1}{\lambda _{1} \lambda _{2}} \Big[ \frac{2}{3} E^{*} + \big( E_{t} - E^{*} \big) \frac{\sigma _{1} \sigma _{2}}{\sigma _{e}^{2}} \Big] , \\ \mathcal{L}_{2222} & =\displaystyle \frac{1}{\lambda _{2}^{2}} \Big[ \frac{4}{3} E^{*} + \big( E_{t} - E^{*} \big) \Big( \frac{\sigma _{2}}{\sigma _{e}} \Big)^{2} - \sigma _{2} \Big] , \end{array} $$

(A.16)

and the shear moduli components are given by:

$$ \textstyle\begin{array}{rl} \mathcal{L}_{1212} & = \displaystyle \frac{1}{\lambda _{2}^{2}} \Big[ \frac{E^{*}}{3} + \frac{\sigma _{2} - \sigma _{1}}{2} \Big] , \\ \mathcal{L}_{2121} & =\displaystyle \frac{1}{\lambda _{1}^{2}} \Big[ \frac{E^{*}}{3} + \frac{\sigma _{1} - \sigma _{2}}{2} \Big] , \\ \mathcal{L}_{1221} = \mathcal{L}_{2112} & = \displaystyle \frac{1}{\lambda _{1} \lambda _{2}} \Big[ \frac{E^{*}}{3} - \frac{\sigma _{1} + \sigma _{2}}{2} \Big] , \end{array} $$

(A.17)

where for the $J_{2}$ flow theory $E^{*} = E$, the $J_{2}$ deformation theory $E^{*} = E_{s}$ and the hyperelastic model $E^{*} = E_{s} [(\lambda _{1}^{2} + \lambda _{2}^{2})/(\lambda _{1}^{2} - \lambda _{2}^{2})](\ln \lambda _{1} - \ln \lambda _{2})$ where these quantities were defined immediately after (A.4). Derivation details for the hyperelastic model can be found in [26]. The principal stresses $\sigma _{\alpha}$ for all three models are identical and given by (A.15).

In the above expressions the material energy density $W$ is the 3D version. We use here in Appendix A the same symbol as for its 2D counterpart in Sect. 2 to avoid extra symbols. In the same spirit, the dimensionless version of the stresses $\sigma _{\alpha}$ and moduli ${\mathcal {L}}_{\alpha \beta \gamma \delta}$ in (A.15), (A.16) and (A.17), are found by taking in all the above expressions the initial Young modulus to be $E=1$.

Appendix B: Influence Discs as Functions of Constitutive Law and Load Orientation

The minimum and maximum influence disc sizes $\delta _{-}$, defined in (2.12), and $\delta _{+}$,defined in (2.13), at the onset of loss of ellipticity $\tau _{m}$, have been presented in [29]. However, for the sake of completeness of the presentation and also in view of a different definition of $\delta _{+}$ adopted in this paper, these results are presented again here in Fig. 13 with the disc sizes given as functions of the load orientation angle $\psi$ for the three different constitutive models considered in Appendix A and for two different power-law hardening exponents, $n=0.22$ (typical of Al alloys, plotted in solid lines) and $n=0.40$ (typical of steel alloys, plotted in dotted lines). Curves in the $\psi < 0$ range are terminated at $\psi =\tan ^{-1}(-0.5)$ below which one stress becomes compressive. Moreover, $J_{2}$ flow theory curves are only plotted for $\psi \le 0$, in view of their well-known – see [24] – unrealistic predictions for $\psi > 0$.

Fig. 13

Minimum $\delta _{-}$ and maximum $\delta _{+}$ influence disc sizes at the onset of loss of ellipticity $\tau _{m}$, as functions of the load orientation angle $\psi$ for the three different constitutive models considered: hyperelastic, $J_{2}$-deformation theory and $J_{2}$-flow theory, all sharing the same uniaxial stress-strain law with $\epsilon _{y} = 0.002$ and $n=0.22$ (solid line) and $n=0.40$ (dotted line)

As expected, for a given material and load orientation $\psi$, both $\delta _{-}$ and $\delta _{+}$ are increasing functions of the hardening exponent $n$. For the lower hardening exponent $n=0.22$, there is practically no difference for either the minimum or maximum influence disc sizes, $\delta _{-}$ and $\delta _{-}$ respectively, between the $J_{2}$ deformation and hyperelastic theory models over the entire range of load orientations considered. For the higher hardening exponent $n=0.4$, the $\delta _{-}$ predictions of the $J_{2}$ deformation and hyperelastic theory models are practically the same over the entire range of loading angles, while the maximum influence disc size $\delta _{+}$ predictions for these two constitutive models start diverging at about $\psi < -\pi /12$.

Setting aside the fact that $\delta _{-}$ and $\delta _{+}$ exist for a significantly smaller range of load orientations for the much stiffer $J_{2}$ flow theory, one can observe that for $n=0.22$ there is no noticeable increase in the $J_{2}$ flow theory value of $\delta _{-}$ compared to its hyperelastic and deformation theory counterparts, but there is a large increase in the corresponding value of $\delta _{+}$, as compared to the other two constitutive models with the same uniaxial curve. The same trend holds true for $n=0.40$, but the differences between constitutive models, especially for $\delta _{+}$, are considerably larger. Moreover, the $J_{2}$ flow theory gives unrealistically high values for $\delta _{+}$, save for a small range near $\psi = 0$.

The small difference of the minimum $\delta _{-}$ and maximum $\delta _{+}$ influence disc sizes predicted by the hyperelastic and $J_{2}$ deformation theory constitutive models is the reason for using the simpler to implement hyperelastic model in most of our numerical calculations.

Appendix C: Critical ($\tau _{m}$) and Stability Limit ($\tau _{L}$) Dimensionless Times

Having established the incremental moduli tensor $\boldsymbol{\mathcal{L}}$ in Appendix A for the constitutive laws adopted in this work, we calculate next the dimensionless critical time $\tau _{m}$ for the onset of loss of ellipticity, the corresponding localization angle $\phi _{m}$ introduced in (2.11) and the dimensionless Lyapunov stability time $\tau _{L}$ in order to establish the range of validity of the plate’s linearized stability, as presented in Sect. 2.2.

From the definition of $\mathbf{A}$ in (2.7) we have

$$ \begin{aligned}[b] & {\mathbf{A}}(\tau , \phi ) \ = \left [ \textstyle\begin{array}{c@{\quad}c} {\mathcal {L}}_{1111}(\tau )(n_{1})^{2} + {\mathcal {L}}_{1212}(\tau )(n_{2})^{2} & ({\mathcal {L}}_{1122}(\tau ) + {\mathcal {L}}_{1221}(\tau ))(n_{1}n_{2}) \\ ({\mathcal {L}}_{2211}(\tau ) + {\mathcal {L}}_{2112}(\tau ))(n_{1}n_{2}) & {\mathcal {L}}_{2121}(\tau )(n_{1})^{2} + {\mathcal {L}}_{2222}(\tau )(n_{2})^{2} \end{array}\displaystyle \right ] \; ; \\ &\quad n_{1}=\cos (\phi ), \; n_{2}=\sin (\phi ) , \end{aligned} $$

(C.1)

and from the definitions in (2.11) and (2.10), $\tau _{m}$ is the lowest positive $\tau$ root of the characteristic equation of $\boldsymbol{A}(\tau , \phi ), \forall \phi \in [0 , \pi /2]$

$$ \begin{aligned}[b] &{\mathcal {L}}_{1111}(\tau ){\mathcal {L}}_{2121}(\tau ) (n_{1})^{4} + [{ \mathcal {L}}_{1111}(\tau ){\mathcal {L}}_{2222}(\tau ) + {\mathcal {L}}_{1212}( \tau ){\mathcal {L}}_{2121}(\tau ) \\ & \quad - ({\mathcal {L}}_{1122}(\tau ){ \mathcal {L}}_{1221}(\tau ))^{2}](n_{1}n_{2})^{2} + {\mathcal {L}}_{1212}( \tau ){\mathcal {L}}_{2222}(\tau ) (n_{2})^{4} . \end{aligned} $$

(C.2)

For the loading angles $\psi$ considered in Sect. 3, we find

$$ \textstyle\begin{array}{ll} \displaystyle \psi = - {\frac{\pi}{12}} & : \quad \begin{aligned}[t] &{}[{\mathcal {L}}_{1111}( \tau _{m}){\mathcal {L}}_{2222}(\tau _{m})]^{1/2} + [{\mathcal {L}}_{1212}( \tau _{m}){\mathcal {L}}_{2121}(\tau _{m})]^{1/2} \\& \quad - [{\mathcal {L}}_{1122}( \tau _{m})+{\mathcal {L}}_{1221}(\tau _{m})] = 0 , \end{aligned} \\ \displaystyle \psi = 0 & : \quad {\mathcal {L}}_{1111}(\tau _{m})= 0 \Longrightarrow \ \tau _{m} = \exp (n)-1 , \\ \displaystyle \psi = {\frac{\pi}{4}} & \displaystyle : \quad { \mathcal {L}}_{1111}(\tau _{m})= 0 \Longrightarrow \ \tau _{m} = \left [\exp ((1 + 3n)/6) -1\right ]\sqrt {2} , \end{array} $$

(C.3)

where the corresponding angles in the interval $[0 , \pi /2]$ are

$$ \begin{aligned}[b] &\displaystyle \psi = - {\frac{\pi}{12}} \ : \ \phi _{m}= \tan ^{-1} \left [{ \frac{{\mathcal {L}}_{1111}(\tau _{m}){\mathcal {L}}_{2121}(\tau _{m})}{{{\mathcal {L}}_{1212}(\tau _{m}){\mathcal {L}}}_{2222}(\tau _{m})}} \right ]^{1/4} , \quad \psi = 0 \ : \ \phi _{m} = 0, \quad \\ &\psi = \frac{\pi}{4} \ : \ \forall \phi _{m} \in \left [0 , \frac{\pi}{2} \right ] . \end{aligned} $$

(C.4)

Notice that for loading angles $\psi = 0, \pi /4$, the critical time is independent of the particular version of the model used (hyperelastic, $J_{2}$ deformation or $J_{2}$ flow theory) and depends solely on the exponent $n$¹⁸ of the power law-type uniaxial stress-strain curve in (A.14).

The Lyapunov (linearized) stability of the system, as discussed in Sect. 2.2, is guaranteed by the positive definiteness of $\boldsymbol{A}(\tau , \phi )$ and $-\dot{\boldsymbol{A}}(\tau , \phi )$. By construction $\boldsymbol{A}(\tau , \phi )$ is positive definite for $\forall \tau \in [0,\tau _{m})$, $\forall \phi \in [0,\pi /2]$. Finding the value of $\tau _{L} \le \tau _{m}$ that guarantees the positive definiteness of $-\dot{\boldsymbol{A}}(\tau , \phi )$ for $\forall \tau \in [0,\tau _{L})$, $\forall \phi \in [0,\pi /2]$ is based on numerical calculations using a symbolic manipulator.

For the cases investigated in Sect. 3 the following values are found for the dimensionless critical time $\tau _{m}$ and the corresponding linearized (Lyapunov) stability time $\tau _{L}$

$$\begin{aligned} {\mathrm{hyperelastic}}\ ; & \ n=0.22 \quad \left \{ \textstyle\begin{array}{l@{\quad}l@{\quad}l} \displaystyle \psi = \tan ^{-1}\left (-{\frac{1}{2}}\right ), & \tau _{m} = 0.508, & \tau _{L} = 0.38 \\ \displaystyle \psi = -{\frac{\pi}{12}}, & \tau _{m} = 0.352, & \tau _{L} = \tau _{m} \\ \displaystyle \psi = 0, & \tau _{m} = 0.246, & \tau _{L} = \tau _{m} \\ \displaystyle \psi = {\frac{\pi}{4}}, & \tau _{m} = 0.451, & \tau _{L} = \tau _{m} \end{array}\displaystyle \right . \\ & \ n=0.40 \quad \left \{ \textstyle\begin{array}{l@{\quad}l@{\quad}l} \displaystyle \psi = \tan ^{-1}\left (-{\frac{1}{2}}\right ), & \tau _{m} = 0.900, & \tau _{L} = 0.33 \\ \displaystyle \psi = -{\frac{\pi}{12}}, & \tau _{m} = 0.715, & \tau _{L} =0.475 \\ \displaystyle \psi = 0, & \tau _{m} = 0.492, & \tau _{L} = \tau _{m} \\ \displaystyle \psi = {\frac{\pi}{4}}, & \tau _{m} = 0.626, & \tau _{L} = \tau _{m} \end{array}\displaystyle \right . \\ J_{2}\ {\mathrm{deformation}}\ ; & \ n=0.22 \quad \left \{ \textstyle\begin{array}{l@{\quad}l@{\quad}l} \displaystyle \psi = \tan ^{-1}\left (-{\frac{1}{2}}\right ), & \tau _{m} = 0.482, & \tau _{L} = 0.44 \\ \displaystyle \psi = -{\frac{\pi}{12}}, & \tau _{m} = 0.351, & \tau _{L} = \tau _{m} \\ \displaystyle \psi = 0, & \tau _{m} = 0.246, & \tau _{L} = \tau _{m} \\ \displaystyle \psi = {\frac{\pi}{4}}, & \tau _{m} = 0.451, & \tau _{L} = \tau _{m} \end{array}\displaystyle \right . \\ & \ n=0.40 \quad \left \{ \textstyle\begin{array}{l@{\quad}l@{\quad}l} \displaystyle \psi = \tan ^{-1}\left (-{\frac{1}{2}}\right ), & \tau _{m} = 0.723, & \tau _{L} = 0.38 \\ \displaystyle \psi = -{\frac{\pi}{12}}, & \tau _{m} = 0.689, & \tau _{L} =0.620 \\ \displaystyle \psi = 0, & \tau _{m} = 0.492, & \tau _{L} = \tau _{m} \\ \displaystyle \psi = {\frac{\pi}{4}}, & \tau _{m} = 0.626, & \tau _{L} = \tau _{m} \end{array}\displaystyle \right . \\ J_{2}\ {\mathrm{flow}}\ ; & \ n=0.22 \quad \left \{ \textstyle\begin{array}{l@{\quad}l@{\quad}l} \displaystyle \psi = \tan ^{-1}\left (-{\frac{1}{2}}\right ), & \tau _{m} = 0.756, & \tau _{L} = 0 \\ \displaystyle \psi = -{\frac{\pi}{12}}, & \tau _{m} = 0.405, & \tau _{L} = 0 \\ \displaystyle \psi = 0, & \tau _{m} = 0.246, & \tau _{L} = 0 \end{array}\displaystyle \right . \\ & \ n=0.40 \quad \left \{ \textstyle\begin{array}{l@{\quad}l@{\quad}l} \displaystyle \psi = \tan ^{-1}\left (-{\frac{1}{2}}\right ), & \tau _{m} = 0.924, & \tau _{L} = 0 \\ \displaystyle \psi = -{\frac{\pi}{12}}, & \tau _{m} = 1.131, & \tau _{L} = 0 \\ \displaystyle \psi = 0, & \tau _{m} = 0.492, & \tau _{L} = 0 \end{array}\displaystyle \right . \end{aligned}$$

(C.5)

It is worth noticing in (C.5) that – for the same hardening exponent $n$ – the results for the hyperelastic and $J_{2}$ deformation theory models are either very close (or coincident for $\psi = 0, \pi /4$), while the much stiffer $J_{2}$ flow theory is not linearly stable since $-\dot{\boldsymbol{A}}(\tau , \phi )$ looses positive definiteness for certain values of $\phi$ at all dimensionless times $\tau$ (and hence $\tau _{L}=0$).