Encyclopedia of Continuum Mechanics

Living Edition
| Editors: Holm Altenbach, Andreas Öchsner

Strain Gradient Plasticity

  • Lorenzo BardellaEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-3-662-53605-6_110-1



Strain gradient plasticity (SGP) is a theory of continuum solid mechanics which aims at modeling the irreversible mechanical behavior of materials, with specific focus on metals and on their response at appropriately small size, typically on the order of micrometers or less. For such small metallic components, a variation in size leads to a peculiar effect, denoted as “smaller being stronger.”


The term plasticity refers to the irreversible mechanical behavior of materials, with particular reference to metals. This behavior occurs when the stress state is large enough for the material to yield, thus leading to a permanent deformation, denoted as plastic deformation. Such deformation can be observed, and the inherent plastic strain measured, after removing a suitable, monotonically applied load which enables yielding. In simple tests, such as uniaxial tension, the yield stress is experimentally recognized as the stress corresponding to the first abrupt change of slope in the stress-strain curve, limiting the elastic (i.e., reversible) regime. The slope of the stress-strain curve after yielding is proportional to the so-called strain hardening. The plastic deformation is mainly due to the nucleation, multiplication, and propagation of dislocations that are line defects within the metal crystal lattice.

SGP is a theory of continuum solid mechanics which aims at modeling the plasticity of metals at appropriately small scale. Specifically, SGP theory focuses on the size range between few tens of nanometers and few tens of micrometers, in which the peculiar “smaller being stronger” size effect has been experimentally measured. The torsion of thin wires (Fleck et al., 1994), the microindentation (Ma and Clarke, 1995), and the microbending (Stölken and Evans, 1998) are among the pioneering experimental observations of such size effect about two decades ago. However, much earlier, Hall (1951) and Petch (1953) discovered the size effect named after them, peculiar of the polycrystalline microstructure of common metals, consisting of many grains (each grain being a single crystal) with randomly distributed lattice orientation. On the basis of Hall and Petch experimental results on metals, such as mild steel and ingot iron, it has been established that the macroscopically observed yield stress increases, with respect to that characterizing the single grain, about proportionally to the inverse of the square root of the average grain size.

The size effect consisting of an increase of yield stress accompanied with diminishing size is referred to as strengthening. Experimental results also show a further size effect, that is, an increase in strain hardening with diminishing size. As demonstrated by Ashby (1970), geometrically necessary dislocations (GNDs) are the main responsible for such size-dependent behavior and, as illustrated in what follows, are related to the gradient of the plastic deformation.

GNDs are also denoted as excess or misfit dislocations in contrast to statistically stored dislocations (SSDs). In fact, in an appropriate average sense, SSDs annihilate each other. The propagation of the whole population of dislocations can be associated with the plastic strain magnitude (Hull and Bacon, 2001). Any single (discrete) dislocation causes a lattice distortion, as illustrated in Fig. 1 for edge and screw dislocations in a cubic crystal. Figure 1 displays the Burgers circuits whose closure failure defines the Burgers vector \(\vec {b}\), characterizing each dislocation (Burgers, 1939), along with the unit vectors \(\vec {m}\), \(\vec {n}\), and \(\vec {t}\) defining the slip direction, the slip plane normal, and the direction orthogonal to \(\vec {m}\) on the slip plane, respectively. The Burgers vector amplitude \(|\vec {b}|\) is equal to an interatomic spacing. Edge dislocations represent half-planes of atoms (black circles in Fig. 1) in an otherwise regular crystal lattice. Hence, the slip direction is normal to the edge dislocation line, while in screw dislocations (Fig. 1 on the right) the slip direction coincides with the dislocation line. Glide is the most relevant component of dislocations’ motion and occurs on a plane containing both the dislocation line and its Burgers vector.
Fig. 1

Schematics for the atomistic characterization of edge (left) and screw (right) dislocations (Figure redrawn and modified from Hayden et al. 1965)

The distortions represented in Fig. 1 are associated with internal stress fields that become very relevant when many dislocations are present, as in metal plasticity. Such stress fields sum up when due to GNDs (which, contrary to SSDs, do not annihilate each other), thus giving rise to long-range stress effects.

In polycrystalline metals subject to plastic deformation, dislocations, locally of the same sign, pile up against grain boundaries, thus forming regions of large GND density referred to as boundary layers.

The size of boundary layers depends on the crystallography and on the grain boundary strength (the larger the latter, the longer the maximum pileup length), while it is not much influenced by the grain size. As a result, the stiffening effect of boundary layers is inversely proportional to the grain size, thus leading to a size effect in the observed strain hardening. Moreover, in a polycrystal, with diminishing grain size, the dislocations’ mean free path decreases along with the possibility of dislocations to enucleate and propagate. Since nucleation and propagation of dislocations have to conspicuously occur at (macroscopic) yield, the above observation offers a qualitative interpretation of the Hall-Petch size effect.

Individual (discrete) dislocations cannot enter a continuum theory, which may instead account for the GND density, that is related to the incompatibility of the plastic distortion field, as shown in the next section.

A strain field is incompatible if it cannot be determined from the gradient of a suitably smooth vector field.

By building on the foregoing concepts, SGP theory extends the conventional plasticity theory (see, e.g., Fleck and Hutchinson, 1997; Gurtin et al., 2010), in such a way that for size above  ≈100 μm, SGP converges to conventional plasticity. At size below a few tens of nanometers, continuum theories may become inappropriate, and the mechanical behavior of metals is dominated by effects neglected by SGP, such as surface effects and dislocation core effects. SGP theory refers to absolute temperature lower than about half of the melting point and to strain rate lower than  ≈10/s (Valdevit and Hutchinson, 2012).


Lightface letters are employed for scalars, whereas boldface letters are used for first-, second-, third-, and fourth-order tensors, respectively represented by small Latin, small Greek, capital Latin, and capital blackboard letters, unless otherwise specified. When index notation is employed, it refers to an orthonormal system of coordinates. The symbol  ⋅  represents the inner product of vectors and tensors (e.g., \(a=\vec {b} \cdot \vec {u} \equiv b_i u_i\), b = σ ⋅ε ≡ σ ij ε ij , \(c=\vec {T} \cdot \vec {S} \equiv T_{ijk} S_{ijk}\)). For any tensor, the modulus reads \(|\boldsymbol {\rho }| \equiv \sqrt {\boldsymbol {\rho } \cdot \boldsymbol {\rho }}\). The symbol  ⊗  denotes the tensor product, e.g., \((\vec {m}\otimes \vec {n})_{ij} \equiv m_i n_j\). The symbol  ×  is adopted for the vector product: \((\vec {t})_i=(\vec {m} \times \vec {n})_i \equiv e_{ijk} m_j n_k=t_i\) and \((\boldsymbol {\zeta } \times \vec {n})_{ij} \equiv e_{jlk} \zeta _{il} n_k\), with e ijk  = (i − j)(j − k)(k − i)/2 denoting the Ricci-Curbastro tensor (or alternating symbol). For the composition of tensors of different orders, the lower-order tensor is on the right, and all its indices are saturated, e.g., \((\vec {t})_i=(\boldsymbol {\sigma } \vec {n})_i \equiv \sigma _{ij} n_j = t_i\), \((\vec {T} \vec {n})_{ij} \equiv T_{ijk} n_k\), and \((\boldsymbol {\sigma })_{ij}=(\mathbb {L}\boldsymbol {\varepsilon })_{ij} \equiv L_{ijkl} \varepsilon _{kl} = \sigma _{ij}\). trγ = γ ⋅δ ≡ γ ii is the trace of a second-order tensor, with δ denoting the second-order identity tensor (or Kronecker symbol). devσ = σ −δ trσ/3, ( symγ) ij  ≡ (γ ij  + γ ji )/2, and ( skwγ) ij  ≡ (γ ij  − γ ji )/2 denote, respectively, the deviatoric, symmetric, and skew-symmetric parts of second-order tensors. By referring to a Cartesian system, (∇ε) ijk  ≡ ∂ε ij /∂x k  ≡ ε ij,k , \((\mbox{div} \vec {S})_{ij} \equiv S_{ijk,k}\), and ( curlγ) ij  ≡ e jkl γ il,k designate, respectively, the gradient, the divergence, and the curl operators. \(\dot {\boldsymbol {\varepsilon }}\) indicates the time derivative dε/dt, with t denoting the variable governing the loading history, not necessarily a physical time.



Attention is restricted to small strains and rotations. In this framework, the gradient of the displacement field \(\vec {u}\) can be additively split into its elastic part, \((\nabla{u})_{el} \), and its plastic part, γ, denoted as the plastic distortion:
$$\displaystyle \begin{aligned} \nabla{u} = (\nabla{u})_{ el}+\boldsymbol{\gamma}\end{aligned} $$
The total strain, plastic strain, elastic strain, and plastic spin are, respectively, defined as
$$\displaystyle \begin{aligned} \begin{array}{rcl} \boldsymbol{\varepsilon} = \mbox{sym}{\nabla{u}}\\ \boldsymbol{\varepsilon}^p = \mbox{sym}{\boldsymbol{\gamma}}\\ \mbox{sym}{(\nabla{u})_{el}} = \boldsymbol{\varepsilon} - \boldsymbol{\varepsilon}^p\\ \boldsymbol{\vartheta}^p = \mbox{skw}{\boldsymbol{\gamma}}\end{array} \end{aligned} $$
2) (3
There exist two main classes of conventional plasticity, that is, crystal plasticity and phenomenological plasticity. The former accounts for the crystal lattice, thus being more precise in the description of the plastic distortion, associated with dislocations’ motion. Instead, phenomenological plasticity neglects the crystal lattice and is mainly employed to model, in an appropriate average sense, the mechanical response of polycrystalline metals, for which the use of crystal plasticity is computationally much more expensive.
More specifically, in crystal plasticity, the plastic distortion is given by the following sum over the slip systems, whose orientation and number A depend on the crystallography (e.g., A = 12 in face-centered cubic metals, such as copper, silver, and gold):
$$\displaystyle \begin{aligned} \boldsymbol{\gamma} = \sum_{\beta = 1}^A \gamma^{(\beta)} \vec{m}^{(\beta)}\otimes \vec{n}^{(\beta)}\end{aligned} $$
in which γ(β), with β = 1, …, A, are the plastic slips and \(\vec {m}^{(\beta )}\) and \(\vec {n}^{(\beta )}\) are the unit vectors defining, respectively, the slip direction and the slip plane normal for the slip system β. Definition (4) implies trγ = 0, meaning that the plastic flow is isochoric. In crystal plasticity, the basic plastic variables are the plastic slips, while this is not the case in phenomenological plasticity, neglecting the crystallography along with Eq. (4).
While the compatibility of the displacement field implies
$$\displaystyle \begin{aligned} \mbox{curl} (\nabla \vec{u}) = \vec{0} \ ,\end{aligned} $$
the incompatibility of γ results, in general, in the closure failure of its circuit C around any suitably smooth surface S within the continuum:
$$\displaystyle \begin{aligned}{b}_{net} = \oint_C \boldsymbol{\gamma} \mathrm{d} {c}\end{aligned} $$
in which \(\vec {c}\) is the vectorial coordinate along C (having pointwise the direction of the tangent to C) and \( {b}_{net}\) is the net Burgers vector, thus mimicking the characterization of a discrete dislocation in the crystal lattice (see Fig. 1). Given that, in general, the surface S is pierced by several dislocations, only dislocations whose Burgers vectors do not cancel out contribute to \({b}_{net}\). Hence, Eq. (6) is associated with the GND density.
By applying Stokes’ theorem, Eq. (6) becomes
$$\displaystyle \begin{aligned} {b}_{net} = \int_S \mbox{curl}\boldsymbol{\gamma}\ {n}_S \mathrm{d} A\end{aligned} $$
where \(\vec {n}_S\) is the unit normal to the surface S, pointing according to the right-hand screw rule, given the positive sense of C.
Equation (7) suggests the definition of Nye’s dislocation density tensor α (Nye, 1953; Kröner, 1962; Fleck and Hutchinson, 1997; Arsenlis and Parks, 1999):
$$\displaystyle \begin{aligned} \boldsymbol{\alpha} = \mbox{curl}\boldsymbol{\gamma}\end{aligned} $$
such that
$$\displaystyle \begin{aligned} {b}_{net} = \int_S \boldsymbol{\alpha} {n}_S \mathrm{d} A\end{aligned} $$
from which one deduces that

Nye’s dislocation density tensor α is a continuum representation of geometrically necessary dislocations such that α ij is the i component of the net Burgers vector related to GNDs of line vector j.

From Eqs. (1), (5), and (8), one finds a link between α and the elastic part of \(\nabla \vec {u}\):
$$\displaystyle \begin{aligned} \boldsymbol{\alpha} = -\mbox{curl} \ (\nabla {u})_{el}\end{aligned} $$
In crystal plasticity, by substituting relation (4) into definition (8), and by making use of the identity e ijk e irs  = δ jr δ ks  − δ js δ kr , one obtains
$$\displaystyle \begin{aligned} \boldsymbol{\alpha} = \sum_{\beta=1}^A \vec{m}^{(\beta)}\otimes\left(\rho_\bot^{(\beta)}\vec{t}^{(\beta)}+\rho_\odot^{(\beta)}\vec{m}^{(\beta)}\right)\end{aligned} $$
in which \(\vec {t}^{(\beta )} = \vec {m}^{(\beta )}\times \vec {n}^{(\beta )}\) and \(\rho _\bot ^{(\beta )} = \nabla \gamma ^{(\beta )}\cdot \vec {m}^{(\beta )}\) and \(\rho _\odot ^{(\beta )} = -\nabla \gamma ^{(\beta )}\cdot \vec {t}^{(\beta )}\) are the projections of the plastic slip gradient onto the slip and transverse directions, respectively. Given the above emphasized property of α, by comparison with the schematics of Fig. 1, one can deduce that ρ⊥(β) and ρ⊙(β) represent, respectively, the densities of pure edge and screw GNDs for the slip system β (Arsenlis and Parks, 1999). Here, the adjective “pure” refers to the fact that the characterization of Fig. 1 is ideal, whereby real dislocations consist of loops, generally having, pointwise, both edge and screw components.

A Quite General Higher-Order Theoretical Framework

Many SGP theories have been developed in literature. While there exist gradient extensions of both phenomenological and crystal plasticity theories (see, e.g., Gurtin et al., 2010 and references therein), here attention is restricted to phenomenological SGP. Moreover, the focus is on the so-called higher-order theories.

Higher-order (HO) theories postulate the existence of HO stresses (often referred to as microstresses) work-conjugate to appropriate combinations of the components of the plastic distortion gradient, ∇γ. Such combinations are then assumed as primal HO kinematic variables, and different choices of them lead to different SGP theories.

HO SGP theories are preferred because they involve HO boundary conditions.

HO boundary conditions are unconventional boundary conditions governing, in the continuum sense, the behavior of dislocations at the boundary. They include the possibility of imposing that dislocations pile up at the boundary, thus forming boundary layers and triggering a nontrivial gradient response even in boundary value problems whose solution would be spatially homogeneous if predicted by a conventional theory.

HO SGP theories refer to the mechanical response of a body occupying a space region Ω, whose external surface ∂Ω, of outward unit normal \(\vec {n}_{\partial \varOmega }\), consists of two couples of complementary parts, such that ∂Ω = ∂Ω s  ∪ ∂Ω u  = ∂Ω f  ∪ ∂Ω h . The conventional tractions \(\vec {s}^0\) are known on ∂Ω s , while the displacement \(\vec {u}^0\) is assigned on ∂Ω u . Dislocations are free to exit the body on ∂Ω f , while dislocations are blocked and may pile up on ∂Ω h .

Most commonly (Dillon and Kratochvíl, 1970; Fleck and Hutchinson, 1997, 2001; Huang et al., 2000; Forest and Sievert, 2003; Gudmundson, 2004; Gurtin, 2004; Polizzotto, 2009; Fleck and Willis, 2009; Gurtin et al., 2010), HO theories are founded on postulating a generalized Principle of Virtual Work (PVW), which requires the appropriate definitions of the internal and external virtual works on any region Π of Ω. Several HO SGP theories may be derived by assuming that the internal virtual work, under the constraints given by relations (2) and (3), reads
$$\displaystyle \begin{aligned} \begin{array}{rcl} {\mathscr{W}}_i(\varPi, \delta\vec{u}, \delta\boldsymbol{\gamma}) &=& \int_\varPi \Big( \boldsymbol{\sigma}\cdot (\delta \boldsymbol{\varepsilon}-\delta\boldsymbol{\varepsilon}^p)\\&+& \boldsymbol{\varsigma}\cdot\delta\boldsymbol{\gamma}+ \vec{S} \cdot \nabla \delta\boldsymbol{\gamma} \Big) \mathrm{d} V\end{array} \end{aligned} $$
in which \(\delta \varepsilon = \dot \varepsilon \delta t\) denotes a compatible variation of the kinematic field ε, σ is the conventional symmetric Cauchy stress, and the existence is then admitted of the unconventional stresses ς and \(\vec {S}\) work-conjugate to γ and ∇γ, respectively. Note that, because of the assumption of isochoric plastic flow, trς = S iij  = 0.
The external virtual work is then provided by the contributions of the volume density of body forces \(\vec {b}_\varPi \) and the contact actions on the boundary of Π, ∂Π, consisting of the two fields \(\vec {s}\) and τ, conjugate to \(\vec {u}\) and γ, respectively:
$$\displaystyle \begin{aligned} \begin{array}{rcl} {\mathscr{W}}_e(\varPi, \delta\vec{u}, \delta\boldsymbol{\gamma}) &=& \int_\varPi \vec{b}_\varPi \cdot \delta \vec{u}\ \mathrm{d} V \\&+& \int_{\partial\varPi} \Big(\vec{s}\cdot \delta \vec{u}+ \boldsymbol{\tau}\cdot \delta\boldsymbol{\gamma} \Big)\mathrm{d} A \\ \end{array} \end{aligned} $$
The generalized PVW equates \({\mathscr {W}}_i\) and \({\mathscr {W}}_e\), as defined in Eqs. (10) and (11). By integrating by parts, using the divergence theorem, and resorting to standard arguments of calculus of variations, one obtains the conventional balance equations:
$$\displaystyle \begin{gathered} \mbox{div} \boldsymbol{\sigma} +\vec{b}_\varPi = \vec{0}\ \ \mbox{in}\ \ \varPi\\ \boldsymbol{\sigma} \vec{n}_{\partial\varPi}=\vec{s} \ \ \mbox{on}\ \ \partial\varPi \end{gathered} $$
12) (13
supplemented by
$$\displaystyle \begin{gathered} \mbox{dev} \boldsymbol{\sigma} - \boldsymbol{\varsigma} + \mbox{div} \vec{S} = \vec{0}\ \ \mbox{in}\ \ \varPi\\ \vec{S} \vec{n}_{\partial\varPi}=\boldsymbol{\tau} \ \ \mbox{on}\ \ \partial\varPi\end{gathered} $$
14) (15
These equations are referred to as higher-order balance equations.

An alternative approach to obtain the governing equations for generalized continua or, more specifically, for SGP theories has been proposed by Del Piero (2009). It consists of postulating the form of the external virtual work only and, then, resorting to indifference requirements and to the Cauchy tetrahedron theorem. Such a procedure allows one to highlight the different role of Eqs. (12) and (13) with respect to Eqs. (14) and (15), thus suggesting to denote the latter as pseudo-balance equations.

SGP theory can alternatively be founded on the so-called insulation condition in a residual-based theory (Polizzotto, 2009), thus avoiding the generalized PVW at all.

Particularization to a Nye’s Tensor-Based SGP and Introduction of Energetic and Dissipative HO Contributions

The following criteria discriminate among different HO SGP theories:
  1. 1.

    the choice of the primal HO kinematic variables (not necessarily the whole ∇γ);

  2. 2.

    whether the HO stress is associated with energetic (recoverable) or dissipative (unrecoverable) processes, or both.

Here, Nye’s dislocation density tensor α is adopted as a primal HO variable because of its physical meaning previously illustrated. However, SGP theories based on different primal HO variables, such as ∇ε p and \(\nabla \dot {\boldsymbol {\varepsilon }}^p\), or even the gradient of the second invariant of \(\dot {\boldsymbol {\varepsilon }}^p\) (in the effort to develop the simplest HO extension of von Mises plasticity), have been successfully proposed in literature (see, e.g., Aifantis, 1984; Zbib and Aifantis, 1992; Fleck and Hutchinson, 2001; Gudmundson, 2004; Fleck and Willis, 2009; Fleck et al., 2015 and references therein). Such theories, with respect to that illustrated next (Gurtin, 2004; Bardella, 2010; Martínez-Pañeda et al., 2016), have the advantage of allowing a simpler implementation.
Here, it is assumed that \(\vec {S}\) admits the decomposition
$$\displaystyle \begin{aligned} {S} = {S}^{(\mbox{def})} + {T}^{(\varepsilon)}\end{aligned} $$
in which
$$\displaystyle \begin{aligned} \begin{array}{rcl} S^{(\mbox{def})}_{ijk} = e_{kjh}\zeta_{ih}-{1\over 3}\delta_{ij}e_{kph}\zeta_{ph}\\ T^{(\varepsilon)}_{ijk} = T^{(\varepsilon)}_{jik}\end{array} \end{aligned} $$
17) (18
where ζ is called the defect stress and definition (17) ensures that ζ is work-conjugate to Nye’s tensor, while \(\vec {T}^{(\varepsilon )}\) is work-conjugate to the plastic strain gradient, because of property (18). Now, it is crucial to point out that there is no redundancy in the choice (16), (17), and (18) because ζ is thought of to be constitutively dependent on the current (total) value of α, thus providing an energetic contribution, while \(\vec {T}^{(\varepsilon )}\) has a dissipative nature, being thought of to be constitutively dependent on the plastic strain gradient rate, \(\nabla \dot {\boldsymbol {\varepsilon }}^p\). In general, the energetic HO stress describes the long-range effect of GNDs at rest, while the dissipative HO stress aims at capturing the irreversibility inherent to GND motion. In order to generalize the conventional flow theory of plasticity, there must be an unconventional dissipative stress work-conjugate to the plastic strain. Here, ς is totally unrecoverable, being constitutively dependent on \(\dot {\boldsymbol {\gamma }}\), as specified later. Adding an energetic stress contribution dependent on ε p (Gudmundson, 2004) would introduce in the theory the conventional kinematic hardening, here neglected. However, the energetic HO stress leads to a “backstress” causing a HO kinematic hardening. Both dissipative and energetic HO contributions are employed because they can describe quite different size effects.

The dissipative HO contribution models the strengthening, in most cases. The size effect predicted by the energetic HO contribution strongly depends on the specific constitutive law adopted; it may consist of an increase in strain hardening with diminishing size, or strengthening, or a combination of both effects.

Hence, by substituting Eqs. (16), (17), and (18) into Eq. (10), the internal virtual work becomes
$$\displaystyle \begin{aligned} {\mathscr{W}}_i(\varPi) = \int_\varPi \Big(\underbrace{\boldsymbol{\sigma}\cdot (\delta \boldsymbol{\varepsilon}-\delta\boldsymbol{\varepsilon}^p) + \boldsymbol{\zeta}\cdot \delta\boldsymbol{\alpha}}_{\mbox{energetic}} + \underbrace{\boldsymbol{\varsigma}\cdot \delta\boldsymbol{\gamma} + \vec{T}^{(\varepsilon)}\cdot \delta\nabla\boldsymbol{\varepsilon}^p}_{\mbox{dissipative}}\Big) \mathrm{d} V\end{aligned} $$
The balance equations along with all the boundary conditions for the whole body can be obtained by standard analytical manipulation on the generalized PVW.
In the absence of conventional body forces and under quasi-static loading, the conventional balance reads
$$\displaystyle \begin{aligned} \begin{array}{rcl} \mbox{div} \boldsymbol{\sigma} = {\mathbf{0}}\ \ \ \mbox{in}\ \ \ \varOmega\\ \boldsymbol{\sigma} \vec{n}_{\partial\varOmega}=\vec{s}^0 \ \mbox{on}\ \partial\varOmega_s \end{array} \end{aligned} $$
19) (20
The HO balance equations are conveniently written by separating the symmetric and skew-symmetric parts of Eqs. (14) and (15). Accordingly, ς is split as
$$\displaystyle \begin{aligned} \begin{array}{rcl} \boldsymbol{\rho} = \mbox{sym}{\boldsymbol{\varsigma}}\\ \boldsymbol{\omega} = \mbox{skw}{\boldsymbol{\varsigma}}\end{array} \end{aligned} $$
thus obtaining
$$\displaystyle \begin{aligned} \begin{array}{rcl} &&\mbox{dev} \boldsymbol{\sigma}-\mbox{sym} [\mbox{dev}(\mbox{curl}{\boldsymbol{\zeta}})] = \boldsymbol{\rho} -\mbox{div}{\vec{T}^{(\varepsilon)}} \ \ \mbox{in}\ \ \varOmega\\ \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl} &&\boldsymbol{\omega}+\mbox{skw} (\mbox{curl}{\boldsymbol{\zeta}}) = {\mathbf{0}} \ \ \ \ \mbox{in}\ \ \varOmega\end{array} \end{aligned} $$
with static boundary conditions:
$$\displaystyle \begin{aligned} \begin{array}{rcl} \vec{T}^{(\varepsilon)}\vec{n}_{\partial\varOmega}+\mbox{sym} [ \mbox{dev}(\boldsymbol{\zeta} \times \vec{n}_{\partial\varOmega})] = {\mathbf{0}} \ \ \ \ \mbox{on}\ \ \partial\varOmega_f\\ \mbox{skw} (\boldsymbol{\zeta} \times \vec{n}_{\partial\varOmega}) = {\mathbf{0}} \ \ \ \ \mbox{on}\ \ \partial\varOmega_f\end{array} \end{aligned} $$

The static HO boundary conditions are homogeneous with the purpose to describe dislocations free to exit the body at ∂Ω f . They are referred to as microfree boundary conditions.

In rate form, the conventional kinematic boundary condition reads
$$\displaystyle \begin{aligned} \dot{\vec{u}} =\dot{\vec{u}}^0\ \ \mbox{on}\ \partial\varOmega_u\end{aligned} $$

Homogeneous HO kinematic boundary conditions are adopted. They are denoted as microhard and describe dislocations piling up at the boundary ∂Ω h .

In the SGP here considered, the form of the microhard conditions depends on whether the dissipative stress \(\vec {T}^{(\varepsilon )}\) is accounted for or not (note that it can be easily neglected by setting to zero a specific material length scale parameter, as specified later). If \(\vec {T}^{(\varepsilon )}\) enters the model, the microhard boundary conditions read
$$\displaystyle \begin{aligned} \dot{\boldsymbol{\varepsilon}}^p=\vec{0}\ \ \mbox{and} \ \ \dot{\boldsymbol{\vartheta}}^p\times \vec{n}_{\partial\varOmega} =\vec{0} \ \ \mbox{on}\ \partial\varOmega_h\end{aligned} $$
Otherwise, if \(\vec {T}^{(\varepsilon )}\) is neglected, one has
$$\displaystyle \begin{aligned} \dot{\boldsymbol{\gamma}}\times \vec{n}_{\partial\varOmega} =\vec{0} \ \ \mbox{on}\ \partial\varOmega_h\end{aligned} $$
Nonhomogeneous boundary conditions may be adopted to model the behavior of polycrystals’ internal grain boundaries, which may become penetrable to (or emit) dislocations when many of them pile up, thus leading to a large internal stress. To this purpose, the jumps of the static and kinematic unconventional variables at the boundary can be constitutively related to the averages of their dual quantities (see, e.g., Gurtin and Needleman, 2005; Fleck and Willis, 2009; Poh and Peerlings, 2016). Such use of the phenomenological SGP here concerned relies on findings on its suitability to provide also reasonable estimates of the behaviour of single crystals (Bardella, 2009, 2010; Poh and Peerlings, 2016).

After substituting the stresses with the kinematic variables through the constitutive laws (as specified next), the HO balance Eqs. (21) and (22) become second-order partial differential equations, representing the yield condition. In particular, in Eq. (21) energetic terms are on the left-hand side in such a way as to highlight the HO backstress contribution given by the defect stress, leading to an unconventional kinematic hardening.

Helmholtz Free Energy Density and Dissipation

The free energy density Ψ depends on both the elastic strain and Nye’s tensor:
$$\displaystyle \begin{aligned} \varPsi(\boldsymbol{\varepsilon}-\boldsymbol{\varepsilon}^{p},\boldsymbol{\alpha}) = \frac{1}{2} \mathbb{L} ( \boldsymbol{\varepsilon}-\boldsymbol{\varepsilon}^{p} ) \cdot ( \boldsymbol{\varepsilon}-\boldsymbol{\varepsilon}^{p} ) + {\mathscr{D}} (\boldsymbol{\alpha})\end{aligned} $$
in which \(\mathbb {L}\) is the elastic stiffness and \({\mathscr {D}} (\boldsymbol {\alpha })\) is the so-called defect energy. Hence, the Cauchy and defect stresses read, respectively
$$\displaystyle \begin{aligned} \begin{array}{rcl} \boldsymbol{\sigma}= \mathbb{L}( \boldsymbol{\varepsilon}-\boldsymbol{\varepsilon}^{p} )\\ \boldsymbol{\zeta} = \frac{\partial \varPsi}{\partial \boldsymbol{\alpha}} = \frac{\partial {\mathscr{D}}(\boldsymbol{\alpha})}{\partial \boldsymbol{\alpha}}\end{array} \end{aligned} $$
For dimensional consistency, \({\mathscr {D}}(\boldsymbol {\alpha })\) must involve at least one material length scale, henceforth referred to as “energetic length scale.” The form of the defect energy is of crucial importance for the modeling.

A one-homogeneous defect energy, e.g., \({\mathscr {D}}(\boldsymbol {\alpha })\propto |\boldsymbol {\alpha }|\), models the strengthening only, while a defect energy quadratic in the whole Nye’s tensor, that is, \({\mathscr {D}}(\boldsymbol {\alpha })\propto |\boldsymbol {\alpha }|{ }^2\), models the increase in strain hardening with diminishing size only, with the energetic length scale governing the boundary layers’ thickness.

The dissipation depends on the following phenomenological effective plastic flow rate extending the definition characterizing conventional von Mises plasticity:
$$\displaystyle \begin{aligned} \dot{E}^p=\sqrt{\frac{2}{3} |\dot{\boldsymbol{\varepsilon}}^p|{}^2+\chi |\dot{\boldsymbol{\vartheta}}^p|{}^2 + \frac{2}{3}L^2 |\nabla \dot{\boldsymbol{\varepsilon}}^p|{}^2 }\end{aligned} $$
in which χ is the material constant governing the dissipation due to the plastic spin and L is a “dissipative” material length scale parameter. \(\dot {E}^p\) is work conjugate to the effective flow resistance
$$\displaystyle \begin{aligned} \varSigma=\sqrt{\frac{3}{2} |\boldsymbol{\rho}|{}^2+\frac{1}{\chi} |\boldsymbol{\omega}|{}^2 + \frac{3}{2L^2} |\vec{T}^{(\varepsilon)}|{}^2 }\end{aligned} $$
under the following definitions for the unrecoverable stresses:
$$\displaystyle \begin{aligned} \begin{array}{rcl} \boldsymbol{\rho}=\frac{2}{3} \frac{\varSigma}{\dot{E}^p} \dot{\boldsymbol{\varepsilon}}^p \\ \boldsymbol{\omega}=\chi \frac{\varSigma}{\dot{E}^p} \dot{\boldsymbol{\vartheta}}^p \\ \vec{T}^{(\varepsilon)}=\frac{2}{3} L^2 \frac{\varSigma}{\dot{E}^p} \nabla \dot{\boldsymbol{\varepsilon}}^p\end{array} \end{aligned} $$
25) (26) (27
This ensures satisfaction of the second law of thermodynamics:
$$\displaystyle \begin{aligned} \boldsymbol{\rho}\cdot\dot{\boldsymbol{\varepsilon}}^p + \boldsymbol{\omega}\cdot\dot{\boldsymbol{\vartheta}}^p + \vec{T}^{(\varepsilon)}\cdot\nabla\dot{\boldsymbol{\varepsilon}}^p \equiv \varSigma\dot E^p > 0 \ \ \ \forall \ \dot{\boldsymbol{\gamma}} \ne \vec{0}\end{aligned} $$
The effective flow resistance is, in general, a function of both \(\dot {E}^p\) and E p , the latter dependence governing the isotropic hardening.
While the strengthening is physically related to the very small amount of plasticity occurring at about the end of what is experimentally recognized as the elastic regime, mathematically, it has been demonstrated that, in the rate-independent case with χ = 0 and in the absence of HO energetic contribution, the constitutive laws (24), (25), (26), and (27) lead to (dissipative) strengthening associated with the loss of stability of the purely elastic state (Chiricotto et al., 2016). Here a viscoplasticity framework is adopted, by properly specifying the dependence of Σ on \(\dot {E}^p\). This allows plasticity to develop at any stress level such that there is no need to implement any yield criterion nor special treatment for the internal evolving boundaries between elastic and plastic regions, the latter being an issue in rate-independent formulations (see, e.g., Fleck and Willis, 2009; Nielsen and Niordson, 2014 and references therein). The effective flow stress is directly given in the form:
$$\displaystyle \begin{aligned} \varSigma(\dot{E}^p,E^p) = \sigma_Y(E^p) V(\dot E^p)\end{aligned} $$
in which σ Y (E p ) is the isotropic hardening law. The following regularization of a unit step function, admitting convex potential, is adopted for \(V(\dot E^p)\), as it allows one to obtain results that are substantially rate-independent, along with computational efficiency (Panteghini and Bardella, 2016):
$$\displaystyle \begin{aligned} V(\dot E^p) = \begin{cases} \displaystyle \frac {\dot E^p}{2 \dot \varepsilon_0} & \mbox{if}\ \ \displaystyle \frac{\dot E^p}{\dot \varepsilon_0}\leq{1} \\ \displaystyle 1-\frac{1}{2} \frac{ \dot \varepsilon_0}{\dot E^p} & \mbox{if}\ \ \displaystyle \frac {\dot E^p}{\dot \varepsilon_0} > 1 \end{cases}\end{aligned} $$
in which \(\dot \varepsilon _0\) is a positive material parameter. Rate independence is obtained for \(\dot \varepsilon _0\to 0\); by suitably approaching this limit, an elastic domain can be numerically observed.

Minimum Principles

Under the assumption that the following dissipation potential exists and is convex in \(\dot {E}^p\)
$$\displaystyle \begin{aligned} {\mathscr{V}} (\dot{E}^p, E^p ) = \int_0^{ \dot{E}^p} \varSigma ( e , E^p ) \mathrm{d} e\end{aligned} $$
along with a convex defect energy, the following minimum principles, useful for computational purposes, hold (Martínez-Pañeda et al., 2016). Under the kinematic constraints (3), (8), and (23), the field \(\dot {\boldsymbol {\gamma }}\) minimizing the functional
$$\displaystyle \begin{aligned} {\mathscr{H}}(\dot{\boldsymbol{\gamma}}) = \int_\varOmega \left[ {\mathscr{V}} (\dot{E}^p, E^p ) + \boldsymbol{\zeta} \cdot \dot{\boldsymbol{\alpha}} - \boldsymbol{\sigma} \cdot \dot{\boldsymbol{\varepsilon}}^p \right]\mathrm{d} V\end{aligned} $$
satisfies the HO balance Eqs. (21) and (22).
Moreover, for a given \(\dot {\boldsymbol {\varepsilon }}^p\), the conventional balance Eqs. (19) and (20) are satisfied by any kinematically admissible field \(\dot {\vec {u}}\) minimizing the functional
$$\displaystyle \begin{aligned} &{\mathscr{J}}(\dot{\vec{u}}) = {1\over 2}\int_\varOmega \mathbb{L}\left( \mbox{sym}\nabla\dot{\vec{u}}-\dot{\boldsymbol{\varepsilon}}^p \right)\\ &\quad\cdot\left( \mbox{sym}\nabla\dot{\vec{u}}-\dot{\boldsymbol{\varepsilon}}^p \right) \mathrm{d} V - \int_{\partial\varOmega_s} \dot{\vec{s}}^0 \cdot \dot{\vec{u}}\ \mathrm{d} A\end{aligned} $$
Minimum principles (29) and (30) extend to the present theory those developed by Fleck and Willis (2009) for a SGP theory adopting the plastic strain gradient as HO primal variable for both the energetic and dissipative contributions.

Example of Application: The Torsion of Thin Metal Wires

The torsion of thin metal wires is an emblematic benchmark for the behavior that SGP aims at modeling. Here, the experimental results of Fleck et al. (1994) are considered.

The wires are constituted by polycrystalline copper and are modeled as homogeneous and isotropic cylinders with circular cross section of radius a. Hence, in cylindrical coordinates, with r, θ, and z denoting, respectively, the radial coordinate, the circumferential coordinate, and the axis of torsion, the displacement field must read
$$\displaystyle \begin{aligned}\begin{array}{rcl} u_{\theta} = \kappa z r\\ u_r = u_z = 0\end{array} \end{aligned} $$
where κ is the applied twist. σ θz is the sole nonvanishing Cauchy stress component, providing the torque T through
$$\displaystyle \begin{aligned} T = 2\pi \int_0^a\sigma_{\theta z}\ r^2 \mathrm{d} r\end{aligned} $$
The sole nonvanishing components of γ and α are \(\varepsilon ^p_{\theta z}(r)\), \(\vartheta ^p_{\theta z}(r) = -\vartheta ^p_{z\theta }(r)\), and
$$\displaystyle \begin{aligned} \begin{array}{rcl} \alpha_{rr} &=& -{\varepsilon^p_{\theta z}+\vartheta^p_{\theta z}\over r}\\ \alpha_{\theta\theta} &=& -{\mathrm{d} \varepsilon^p_{\theta z}\over \mathrm{d} r}-{\mathrm{d} \vartheta^p_{\theta z} \over \mathrm{d} r}\\ \alpha_{zz} &=& {\varepsilon^p_{\theta z}-\vartheta^p_{\theta z}\over r}+ {\mathrm{d} \varepsilon^p_{\theta z}\over \mathrm{d} r}-{\mathrm{d} \vartheta^p_{\theta z} \over \mathrm{d} r}\end{array} \end{aligned} $$
All these components represent densities of pure screw dislocations and suggest to include in the defect energy a dependence on the invariant trα, which is a function of the plastic spin only, along with the essential dependence on | devα|.
The following regularization of the defect energy proposed by Forest and Guéninchault (2013) is considered (see also Groma et al., 2007; Svendsen and Bargmann, 2010):
$$\displaystyle \begin{aligned} \begin{array}{rcl} {\mathscr{D}}(\boldsymbol{\alpha}) &=& \mu \ell_1|\mbox{tr} \boldsymbol{\alpha}| \ln\left({\ell_1|\mbox{tr} \boldsymbol{\alpha}|\over \alpha_1} +1 \right) \\&+& \mu \ell_2|\mbox{dev} \boldsymbol{\alpha}| \ln\left({\ell_2|\mbox{dev} \boldsymbol{\alpha}|\over \alpha_2} +1 \right)\end{array} \end{aligned} $$
with 1 and 2 denoting independent energetic material length scales and α 1 and α 2 further positive material parameters.
A power law is adopted for the isotropic hardening:
$$\displaystyle \begin{aligned} \sigma_Y(E^p) = \sigma_0+H (E^p)^{N_h}\end{aligned} $$
in which σ 0 is the initial yield stress and H and N h are nonnegative material parameters.

In the torsion problem, dislocations pile up at the wire center, where \(\varepsilon ^p_{\theta z}(0)=\vartheta ^p_{\theta z}(0) = 0\), while they are free to exit the wire at r = a, where microfree conditions are imposed. The results reported next have been obtained by Bardella and Panteghini (2015) by an ad hoc implicit finite element implementation of this problem.

Figure 2 shows the comparison between the experimental results and the theoretical predictions, in terms of the normalized torque T/a 3 as a function of the nondimensional twist κa, that is, the maximum deformation experienced by the wire for a given κ. If there were no size effects, the theoretical curves in Fig. 2 would superimpose exactly, as well as the experimental ones in the absence of uncertainty and fluctuations. Even by neglecting the HO dissipation (i.e., by setting L =  0) and without resorting to any specific identification procedure, the following material parameters lead to a quite good prediction of the experimentally observed size effect: shear modulus μ =  45 GPa, σ 0 =  68 MPa, H =  150 MPa, N h  =  0.37, \(\dot \varepsilon _0=\) 1.E-5 s−1, 1 =  2.1E-4 μm, 2 =  2.1E-3 μm, α 1 ≈ 2.8571E-4, and α 2 ≈ 5.7143E-3, χ =  2/3. Note that a finite deformation framework would be more appropriate in order to predict the experimental results reported in Fig. 2.
Fig. 2

Comparison with the experimental results of Fleck et al. (1994); void symbols represent the theoretical results (Figure adapted from Bardella and Panteghini 2015)

Open Problems

Predictions Under Nonproportional Loading in the Presence of HO Dissipative Contributions

The SGP here considered is of the non-incremental type, in the terminology of Fleck et al. (2014), referring to the constitutive laws governing the dissipation. In fact, in non-incremental theories, a finite HO stress is constitutively related to the rate of the chosen primal HO kinematic variable. In this case, rate-independent SGP may lead to an incremental purely elastic response, referred to as “elastic gap,” when changing the loading direction after having conspicuously developed plasticity under proportional loading (Fleck et al., 2014, 2015; Bardella and Panteghini, 2015; Fleck and Willis, 2015; Carstensen et al., 2017). Whether this is a physical behavior or not should be discerned by suitable experiments. Non-incremental SGP theories are employed because their framework makes it easy to satisfy the second law of thermodynamics, as in Eq. (28).

In incremental SGP theories (see, e.g., Fleck and Hutchinson, 2001), instead, the rate of the HO dissipative stress is constitutively related to the chosen primal HO kinematic variable. This class of theories is, on the one hand, free from “elastic gap” under nonproportional loading (Fleck et al., 2014). On the other hand, in such theories, it is difficult to ensure the satisfaction of the second law of thermodynamics for arbitrary loading history (Gudmundson, 2004; Gurtin and Anand, 2009; Fleck et al., 2015).

Cyclic Behavior Utilizing a Defect Energy that Predicts Strengthening

By referring to α as primal HO variable, a defect energy allowing the prediction of conspicuous (energetic) strengthening has to be such that at very low |α| a small increase of |α| provides a large increase of |ζ| and, then, a much slower increase of |ζ| with |α| for larger values of |α|. This is the case of the logarithmic form (31) or, at the largest extent, of the one-homogeneous form \({\mathscr {D}} = \ell \mu |\boldsymbol {\alpha }|\). Under cyclic loading, this turns out in a stress-strain curve that becomes concave at a certain point after inverting the load. This has been explained by resorting to the observation that the last dislocation piling up is the first leaving the pileup when inverting the load. Even though this explanation is perfectly appropriate for strain gradient crystal plasticity under single slip (Wulfinghoff et al., 2015), there is the need of further investigations in multislip and, most of all, in polycrystalline plasticity.



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Copyright information

© Springer-Verlag GmbH Germany 2018

Authors and Affiliations

  1. 1.Department of Civil, Environmental, Architectural Engineering and MathematicsUniversity of BresciaBresciaItaly

Section editors and affiliations

  • Samuel Forest
    • 1
  1. 1.Centre des Matériaux UMR 7633Mines ParisTech CNRSEvry CedexFrance