# Strain Gradient Plasticity

**DOI:**https://doi.org/10.1007/978-3-662-53605-6_110-1

## Synonyms

## Definitions

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.”

## Background

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.

*Glide*is the most relevant component of dislocations’ motion and occurs on a plane containing both the dislocation line and its Burgers vector.

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).

### Notation

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 \((\boldsymbol {\sigma })_{ij}=(\mathbb {L}\boldsymbol {\varepsilon })_{ij} \equiv L_{ijkl} \varepsilon _{kl} = \sigma _{ij}\). tr

*=*

**γ***⋅*

**γ***≡*

**δ***γ*

_{ ii }is the trace of a second-order tensor, with

*denoting the second-order identity tensor (or Kronecker symbol). dev*

**δ***=*

**σ***−*

**σ***tr*

**δ***/3, ( sym*

**σ***)*

**γ**_{ ij }≡ (

*γ*

_{ ij }+

*γ*

_{ ji })/2, and ( skw

*)*

**γ**_{ 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 ( curl

*)*

**γ**_{ ij }≡

*e*

_{ jkl }

*γ*

_{ il,k }designate, respectively, the gradient, the divergence, and the curl operators. \(\dot {\boldsymbol {\varepsilon }}\) indicates the time derivative d

*/d*

**ε***t*, with

*t*denoting the variable governing the loading history, not necessarily a physical time.

## Theory

### Kinematics

*, denoted as the plastic distortion:*

**γ***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.

*A*depend on the crystallography (e.g.,

*A*= 12 in face-centered cubic metals, such as copper, silver, and gold):

*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 tr

*= 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).*

**γ***results, in general, in the closure failure of its circuit*

**γ***C*around any suitably smooth surface

*S*within the continuum:

*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.

*S*, pointing according to the right-hand screw rule, given the positive sense of

*C*.

*(Nye, 1953; Kröner, 1962; Fleck and Hutchinson, 1997; Arsenlis and Parks, 1999):*

**α**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*.

*and the elastic part of \(\nabla \vec {u}\):*

**α***e*

_{ ijk }

*e*

_{ irs }=

*δ*

_{ jr }

*δ*

_{ ks }−

*δ*

_{ js }

*δ*

_{ kr }, one obtains

*, by comparison with the schematics of Fig. 1, one can deduce that*

**α***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 }.

*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

*ε*,

*is the conventional symmetric Cauchy stress, and the existence is then admitted of the unconventional stresses*

**σ***and \(\vec {S}\) work-conjugate to*

**ς***and ∇*

**γ***, respectively. Note that, because of the assumption of isochoric plastic flow, tr*

**γ***=*

**ς***S*

_{ iij }= 0.

*Π*,

*∂Π*, consisting of the two fields \(\vec {s}\) and

*, conjugate to \(\vec {u}\) and*

**τ***, respectively:*

**γ**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

- 1.
the choice of the primal HO kinematic variables (not necessarily the whole ∇

);**γ** - 2.
whether the HO stress is associated with energetic (recoverable) or dissipative (unrecoverable) processes,

*or both*.

*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 ∇*

**α**

**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.

*is called the*

**ζ***defect stress*and definition (17) ensures that

*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*

**ζ***, 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*

**α***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,

*is totally unrecoverable, being constitutively dependent on \(\dot {\boldsymbol {\gamma }}\), as specified later. Adding an energetic stress contribution dependent on*

**ς**

**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.

*for the whole body*can be obtained by standard analytical manipulation on the generalized PVW.

*is split as*

**ς**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.

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

*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

*Ψ*depends on both the elastic strain and Nye’s tensor:

*defect energy*. Hence, the Cauchy and defect stresses read, respectively

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.

*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

*E*

^{ p }, the latter dependence governing the isotropic hardening.

*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:

*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):

#### Minimum Principles

### 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.

*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

*κ*is the applied twist.

*σ*

_{ θz }is the sole nonvanishing Cauchy stress component, providing the torque

*T*through

*and*

**γ***are \(\varepsilon ^p_{\theta z}(r)\), \(\vartheta ^p_{\theta z}(r) = -\vartheta ^p_{z\theta }(r)\), and*

**α***, which is a function of the plastic spin only, along with the essential dependence on | dev*

**α***|.*

**α***ℓ*

_{1}and

*ℓ*

_{2}denoting independent energetic material length scales and

*α*

_{1}and

*α*

_{2}further positive material parameters.

*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.

*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},

*ℓ*

_{1}= 2.1E-4 μm,

*ℓ*

_{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.

## 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 |

*| a small increase of |*

**α***| provides a large increase of |*

**α***| and, then, a much slower increase of |*

**ζ***| with |*

**ζ***| 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.*

**α**## Cross-References

## References

- Aifantis EC (1984) On the microstructural origin of certain inelastic models. J Eng Mater Tech-T ASME 106:326–330CrossRefGoogle Scholar
- Arsenlis A, Parks DM (1999) Crystallographic aspects of geometrically-necessary and statistically-stored dislocation density. Acta Mater 47:1597–1611CrossRefGoogle Scholar
- Ashby MF (1970) The deformation of plastically non-homogeneous materials. Philos Mag 21:399–424CrossRefGoogle Scholar
- Bardella L (2009) A comparison between crystal and isotropic strain gradient plasticity theories with accent on the role of the plastic spin. Eur J Mech A-Solid 28:638–646CrossRefzbMATHGoogle Scholar
- Bardella L (2010) Size effects in phenomenological strain gradient plasticity constitutively involving the plastic spin. Int J Eng Sci 48:550–568MathSciNetCrossRefzbMATHGoogle Scholar
- Bardella L, Panteghini A (2015) Modelling the torsion of thin metal wires by distortion gradient plasticity. J Mech Phys Solids 78:467–492MathSciNetCrossRefzbMATHGoogle Scholar
- Burgers JM (1939) Some considerations of the field of stress connected with dislocations in a regular crystal lattice. K Ned Akad Van Wet 42:293–325 (Part 1), 378–399 (Part 2)Google Scholar
- Carstensen C, Ebobisse F, McBride AT, Reddy BD, Steinmann P (2017) Some properties of the dissipative model of strain-gradient plasticity. Philos Mag 97: 693–717CrossRefGoogle Scholar
- Chiricotto M, Giacometti L, Tomassetti G (2016) Dissipative scale effects in strain-gradient plasticity: the case of simple shear. SIAM J Appl Math 76:688–704MathSciNetCrossRefzbMATHGoogle Scholar
- Del Piero G (2009) On the method of virtual power in continuum mechanics. J Mech Mater Struct 4:281–292CrossRefGoogle Scholar
- Dillon OW J, Kratochvíl J (1970) A strain gradient theory of plasticity. Int J Solids Struct 6:1513–1533CrossRefzbMATHGoogle Scholar
- Fleck NA, Hutchinson JW (1997) Strain gradient plasticity. Adv Appl Mech 33:295–361CrossRefzbMATHGoogle Scholar
- Fleck NA, Hutchinson JW (2001) A reformulation of strain gradient plasticity. J Mech Phys Solids 49:2245–2271CrossRefzbMATHGoogle Scholar
- Fleck NA, Willis JR (2009) A mathematical basis for strain-gradient plasticity theory. Part II: tensorial plastic multiplier. J Mech Phys Solids 57:1045–1057zbMATHGoogle Scholar
- Fleck NA, Willis JR (2015) Strain gradient plasticity: energetic or dissipative? Acta Mech Sinica 31:465–472MathSciNetCrossRefzbMATHGoogle Scholar
- Fleck NA, Muller GM, Ashby MF, Hutchinson JW (1994) Strain gradient plasticity: theory and experiments. Acta Metall Mater 42:475–487CrossRefGoogle Scholar
- Fleck NA, Hutchinson JW, Willis JR (2014) Strain gradient plasticity under non-proportional loading. Proc R Soc Lond A 470:20140267CrossRefGoogle Scholar
- Fleck NA, Hutchinson JW, Willis JR (2015) Guidelines for constructing strain gradient plasticity theories. J Appl Mech-T ASME 82:1–10CrossRefGoogle Scholar
- Forest S, Guéninchault N (2013) Inspection of free energy functions in gradient crystal plasticity. Acta Mech Sinica 29:763–772MathSciNetCrossRefzbMATHGoogle Scholar
- Forest S, Sievert R (2003) Elastoviscoplastic constitutive frameworks for generalized continua. Acta Mech 160:71–111CrossRefzbMATHGoogle Scholar
- Groma I, Györgyi G, Kocsis B (2007) Dynamics of coarse grained dislocation densities from an effective free energy. Philos Mag 87:1185–1199CrossRefGoogle Scholar
- Gudmundson P (2004) A unified treatment of strain gradient plasticity. J Mech Phys Solids 52:1379–1406MathSciNetCrossRefzbMATHGoogle Scholar
- Gurtin ME (2004) A gradient theory of small-deformation isotropic plasticity that accounts for the Burgers vector and for dissipation due to plastic spin. J Mech Phys Solids 52:2545–2568MathSciNetCrossRefzbMATHGoogle Scholar
- Gurtin ME, Anand L (2009) Thermodynamics applied to gradient theories involving the accumulated plastic strain: the theories of Aifantis and Fleck & Hutchinson and their generalization. J Mech Phys Solids 57: 405–421MathSciNetCrossRefzbMATHGoogle Scholar
- Gurtin ME, Needleman A (2005) Boundary conditions in small-deformation, single-crystal plasticity that account for the Burgers vector. J Mech Phys Solids 53: 1–31MathSciNetCrossRefzbMATHGoogle Scholar
- Gurtin ME, Fried E, Anand L (2010) The mechanics and thermodynamics of continua. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Hall EO (1951) The deformation and ageing of mild steel: III discussion of results. Proc Phys Soc B 64:747–753CrossRefGoogle Scholar
- Hayden W, Moffatt WG, Wulff J (1965) The structure and properties of materials: vol III, mechanical behavior. Wiley, New YorkGoogle Scholar
- Huang Y, Gao H, Nix WD, Hutchinson JW (2000) Mechanism-based strain gradient plasticity – II. Analysis. J Mech Phys Solids 48:99–128MathSciNetCrossRefzbMATHGoogle Scholar
- Hull D, Bacon DJ (2001) Introduction to dislocations, 4th edn. Butterworth-Heinemann, OxfordGoogle Scholar
- Kröner E (1962) Dislocations and continuum mechanics. Appl Mech Rev 15:599–606Google Scholar
- Ma Q, Clarke DR (1995) Size dependent hardness in silver single crystals. J Mater Res 10:853–863CrossRefGoogle Scholar
- Martínez-Pañeda E, Niordson CF, Bardella L (2016) A finite element framework for distortion gradient plasticity with applications to bending of thin foils. Int J Solids Struct 96:288–299CrossRefGoogle Scholar
- Nielsen KL, Niordson CF (2014) A numerical basis for strain-gradient plasticity theory: rate-independent and rate-dependent formulations. J Mech Phys Solids 63:113–127MathSciNetCrossRefzbMATHGoogle Scholar
- Nye JF (1953) Some geometrical relations in dislocated crystals. Acta Metall 1:153–162CrossRefGoogle Scholar
- Panteghini A, Bardella L (2016) On the finite element implementation of higher-order gradient plasticity, with focus on theories based on plastic distortion incompatibility. Comput Method Appl M 310:840–865MathSciNetCrossRefGoogle Scholar
- Petch NJ (1953) The cleavage strength of polycrystals. J Iron Steel Inst 174:25–28Google Scholar
- Poh LH, Peerlings RHJ (2016) The plastic rotation effect in an isotropic gradient plasticity model for applications at the meso scale. Int J Solids Struct 78–79:57–69CrossRefGoogle Scholar
- Polizzotto C (2009) A link between the residual-based gradient plasticity theory and the analogous theories based on the virtual work principle. Int J Plasticity 25:2169–2180CrossRefGoogle Scholar
- Stölken JS, Evans AG (1998) A microbend test method for measuring the plasticity length scale. Acta Mater 46:5109–5115CrossRefGoogle Scholar
- Svendsen B, Bargmann S (2010) On the continuum thermodynamic rate variational formulation of models for extended crystal plasticity at large deformation. J Mech Phys Solids 58:1253–1271MathSciNetCrossRefzbMATHGoogle Scholar
- Valdevit L, Hutchinson JW (2012) Plasticity theory at small scales. In: Bhushan B (ed) Encyclopedia of nanotechnology. Springer, Dordrecht, pp 3319–3327Google Scholar
- Wulfinghoff S, Forest S, Böhlke T (2015) Strain gradient plasticity modelling of the cyclic behaviour of laminate microstructures. J Mech Phys Solids 79:1–20MathSciNetCrossRefzbMATHGoogle Scholar
- Zbib HM, Aifantis EC (1992) On the gradient-dependent theory of plasticity and shear banding. Acta Mech 92:209–225CrossRefzbMATHGoogle Scholar