A review of higher order strain gradient theories of plasticity: Origins, thermodynamics and connections with dislocation mechanics

Abstract

In this paper we review developments in higher order strain gradient theories. Several variants of these theories have been proposed in order to explain the effects of size on plastic properties that are manifest in several experiments with micron sized metallic structures. It is generally appreciated that the size effect arises from the storage of geometrically necessary dislocations (GNDs) over and above the statistically stored dislocations (SSDs) required for homogeneous deformations. We review developments that show that the GNDs result from the non-homogeneous nature of the deformation field. Though the connection between GNDs and strain gradients are established in the framework of single crystal plasticity, generalisations to polycrystal plasticity has been made. Strain gradient plasticity inherently involves an intrinsic length scale. In our review, we show, through a few illustrative problems, that conventional plasticity solutions can always be reduced to a scale independent form. The same problems are solved with a simple higher order strain gradient formulation to capture the experimentally observed size effects. However, higher order theories need to be thermodynamically consistent. It has recently been shown that only a few of the existing theories pass this test. We review a few that do. Higher order theories require higher order boundary conditions that enable us to model effects of dislocation storage at impermeable boundaries. But these additional boundary conditions also lead to unique conceptual issues that are not encountered in conventional theories. We review attempts at resolving these issues pertaining to higher order boundary conditions. Finally, we review the future of such theories, their relevance and experimental validation.

Appendix A

A rate independent constitutive relation for an incremental formulation is given by Fleck & Hutchinson (2001) where

$$ \dot \tau^{p} = h(E^{p})\dot\epsilon^{p}, \;\; \text{and} \;\; \dot\xi_{i} = l_{*}^{2}h(E^{p})\dot\epsilon^{p}_{,i}. $$

(141)

Here h(E ^p)=d σ _c /d E ^p is the hardening modulus evaluated at E ^p. Replacing the constitutive relations in the microforce balance $\tau ^{p}=\sigma _{e} + \xi ^{p}_{i,i}$, results in a flow rule which is a second order differential equation in terms of 𝜖 ^p. The flow rule thus obtained is essentially a variant of the one given in Eq. (109):

$$ \dot\sigma_{e} = h\dot\epsilon^{p} - l_{*}^{2}h\nabla^{2}\dot\epsilon^{p}. $$

(142)

In deriving the above, the spatial variation of h has been ignored. However, it has been argued (Morton E Gurtin & Lallit Anand 2009; Peter Gudmundson 2004) that the above constitutive relations are not consistent with the second law of the thermodynamics for certain strain loading paths. In fact, Morton E Gurtin & Lallit Anand (2009) have shown that for a particular loading path the above relations fail to satisfy the dissipation inequality (Eq. (88)). A simplified proof of this fact is presented here.

Following (Morton E Gurtin & Lallit Anand 2009) we start with a simpler form of the constitutive equations given in Eq. (141) namely,

$$ \dot \tau^{p}=H(\epsilon^{p})\dot\epsilon^{p}, \;\; \text{and} \;\; \dot\xi_{k}=G(\epsilon^{p})\dot\epsilon^{p}_{,k}. $$

(143)

Incidentally, the violation of the dissipation inequality can also be proved for the constitutive model in Eq. (141) but is substantially more tedious. Additionally we assume that,

$$\begin{array}{@{}rcl@{}} &&\epsilon^{p}=\epsilon^{p}_{,k}= 0, \;\; \text{at} \;\; t=0, \;\; \text{and} \\ &&\dot\epsilon^{p} \ge 0, \;\; \text{for} \;\; t \ge 0. \end{array} $$

(144)

By integrating the above constitutive relations, consistent with the above initial conditions one obtains,

$$ \tau^{p}=S(\epsilon^{p}) \;\; \text{and} \;\; \xi_{k} = {\int\limits_{0}^{t}} G(\epsilon^{p})\epsilon^{p}_{,k}dt, $$

(145)

with an assumption that τ ^p(0)=0 and ξ _k (0)=0. As $\bar {\boldsymbol {\xi }}^{p}=\boldsymbol {0}$ for rate independent models, the higher order stress

$$ {\xi^{p}_{k}} = \frac{\partial\psi^{p}}{\partial\epsilon^{p}_{,k}}, $$

(146)

which, from Eq. (145), can be written as

$$ {\xi^{p}_{k}}= \left. \frac{\partial\psi^{p}(\epsilon^{p},\epsilon^{p}_{,k})}{\partial\epsilon^{p}_{,k}} \right\vert_{t} ={\int\limits_{0}^{t}} G(\epsilon^{p}(y))\dot\epsilon^{p}_{,k}(y)dy. $$

(147)

Since for no strain history should we detect a possible violation of the dissipation inequality, we choose a strain history (shown in figure 13) of the form

$$\begin{array}{@{}rcl@{}} &&\epsilon^{p}_{\delta}(y) = 0, \;\; \text{for} \;\; 0 \le y \le t-\delta \\ &&\epsilon^{p}_{\delta}(t) = \epsilon^{p}(t). \end{array} $$

(148)

Under this history, the right side of Eq. (147) becomes

$$ {\int\limits_{0}^{t}} G(\epsilon^{p}_{\delta}(y))\dot\epsilon^{p}_{,k}(y) dy = \left[ G(0)\epsilon^{p}_{,k}(y)\right]_{0}^{t-\delta} + \int\limits_{t-\delta}^{t} G(\epsilon^{p}_{\delta}(y))\dot\epsilon^{p}_{,k}(y). $$

(149)

For a limiting case when δ→0, it can be observed using Eq. (144) that,

$$ {\int\limits_{0}^{t}} G(\epsilon^{p}_{\delta}(y))\dot\epsilon^{p}_{,k}(y) dy = G(0)\epsilon^{p}_{,k}(t) = {\int\limits_{0}^{t}} G(0)\dot\epsilon^{p}_{,k}(y) dy $$

(150)

leading to

$${\int\limits_{0}^{t}} (G(\epsilon^{p}_{\delta}(y)) - G(0) )\dot\epsilon^{p}_{,k} dy = 0. $$

(151)

The above relation is true for any $\dot \epsilon ^{p}_{,k}(y)$ and hence $G(\epsilon ^{p}_{\delta }(y)) = G(0) = G$ must be a constant, independent of 𝜖 ^p. For G= constant, Eq. (147) after integration gives a free energy of the form,

$$ \psi^{p}(\epsilon^{p},\epsilon^{p}_{,k}) = \frac{1}{2}G\epsilon^{p}_{,k}\epsilon^{p}_{,k} + \phi(\epsilon^{p}). $$

(152)

Consequently the flow rule becomes,

$$ \sigma_{e}=S(\epsilon^{p}) - G\nabla^{2}\epsilon^{p}, $$

(153)

which is exactly same as the flow rule due to Aifantis (1984) in Eq. (109). As shown in Eq. (153) this form can only result when the higher order stress is completely energetic in nature. It can also be noticed that a vanishing free energy (G=0) makes the above relation to reduce to a flow rule

$$ \sigma_{e} = S(\epsilon^{p}), $$

(154)

conventional in nature.

Figure 13

Schematic of the strain history given in Eq. (148).