The bimodal theory of plasticity II: polar material response

Abstract

Based on arguments of the invariant theory of tensor representations, the first part of this study [Soldatos, J Mech Phys Solids 59:1576–1595, 2011] generalised the experimentally established theory of bimodal plasticity in a manner which elucidates its physical and mathematical connection with the plasticity theory of ideal fibre-reinforced solids. The present, second, part of the study deals with a major relevant question which remains unanswered and refers to the considerable deviation which the plastic-strain vector has shown in experiments from the normal to the yield surface direction. The described flow rule discrepancy is resolved here by reconsidering the standard assumption of perfectly flexible fibres. All previously established experimental and theoretical results and observations which refer to bimodal plasticity remain intact and guide this investigation, which, however, postulates further that strong fibres possess bending stiffness. In turn, fibre resistance in bending implies the presence of couple stress and non-symmetric stress, and as a result, the plastic response of the fibrous composite acquires features of polar material behaviour. The resulting polar material version of the bimodal plasticity model provides a convincing explanation, as well as the reasons behind the failure of symmetric-stress (non-polar material) plasticity to conform to well-known and well-established principles of plastic flow. It is thus seen that if polar plastic behaviour is erroneously described with arguments of symmetric-stress plasticity, the corresponding experimental or theoretical measurements become relevant to a scaled rather than the actual yield surface. Those two surfaces resemble each other closely, but, in agreement with existing experimental evidence, their normal directions are generally different. Experimental verification of this kind of polar plastic behaviour may accordingly become possible by measuring simultaneously the values of shear-stress components acting along as well as across fibres throughout the loading process of a relevant fibrous composite specimen.

Appendix: Decomposition of plastic deformation-rate tensor (8.2)

The non-symmetric plastic deformation-rate tensor (8.2) may be decomposed into symmetric and anti-symmetric parts as follows:

$$\begin{aligned} q_{\alpha \beta }^p =d_{\alpha \beta }^p +\omega _{\alpha \beta }^p , \end{aligned}$$

(10.1)

where, according to the relevant standard decomposition rule,

$$\begin{aligned} \begin{aligned} d_{\alpha \beta }^p&= q_{\left( {\alpha \beta } \right) }^p =\frac{1}{2}\left( {q_{\alpha \beta }^p +q_{\beta \alpha }^p } \right) =\frac{1}{2}\dot{\lambda }\left( {\frac{\partial f}{\partial \sigma _{\alpha \beta } }+\frac{\partial f}{\partial \sigma _{\beta a} }} \right) , \\ \omega _{\alpha \beta }^p&= q_{\left[ {\alpha \beta } \right] }^p =\frac{1}{2}\left( {q_{\alpha \beta }^p -q_{\beta \alpha }^p } \right) =\frac{1}{2}\dot{\lambda }\left( {\frac{\partial f}{\partial \sigma _{\alpha \beta } }-\frac{\partial f}{\partial \sigma _{\beta a} }} \right) . \end{aligned} \end{aligned}$$

(10.2)

Nevertheless, the use of (2.1) yields

$$\begin{aligned} \frac{\partial f}{\partial \sigma _{\alpha \beta } }&= \frac{\partial f}{\partial \sigma _{\left( {\mu \nu } \right) } }\frac{\partial \sigma _{\left( {\mu \nu } \right) } }{\partial \sigma _{\alpha \beta } }+\frac{\partial f}{\partial \sigma _{\left[ {\mu \nu } \right] } }\frac{\partial \sigma _{\left[ {\mu \nu } \right] } }{\partial \sigma _{\alpha \beta } } \nonumber \\&= \frac{\partial f}{\partial \sigma _{\left( {\mu \nu } \right) } }\frac{1}{2}\left( {\delta _{\alpha \mu } \delta _{\beta \nu } +\delta _{\alpha \nu } \delta _{\beta \mu } } \right) +\frac{\partial f}{\partial \sigma _{\left[ {\mu \nu } \right] } }\frac{1}{2}\left( {\delta _{\alpha \mu } \delta _{\beta \nu } -\delta _{\alpha \nu } \delta _{\beta \mu } } \right) \nonumber \\&= \frac{1}{2}\left( {\frac{\partial f}{\partial \sigma _{\left( {\alpha \beta } \right) } }+\frac{\partial f}{\partial \sigma _{\left( {\beta \alpha } \right) } }} \right) +\frac{1}{2}\left( {\frac{\partial f}{\partial \sigma _{\left[ {\alpha \beta } \right] } }-\frac{\partial f}{\partial \sigma _{\left[ {\beta \alpha } \right] } }} \right) =\frac{\partial f}{\partial \sigma _{\left( {\alpha \beta } \right) } }+\frac{\partial f}{\partial \sigma _{\left[ {\beta \alpha } \right] } }, \end{aligned}$$

(10.3)

where the relation

$$\begin{aligned} \frac{\partial f}{\partial \sigma _{\left[ {\beta \alpha } \right] } }=\frac{\partial f}{\partial \left( {-\sigma _{\left[ {\alpha \beta } \right] } } \right) }=-\frac{\partial f}{\partial \sigma _{\left[ {\alpha \beta } \right] }} \end{aligned}$$

(10.4)

has also been used. Introduction of the result (10.3) to the right-hand side of (10.2) followed by the use of (10.4) then leads to the definition (8.3) of the plastic strain rate and plastic spin tensors.