Crystal plasticity

Pull on a single crystal of copper, and the stress at which it yields depends on how the crystal was cut. Along one direction it flows at half the stress that another direction needs. An isotropic yield criterion, with one yield stress for every direction, cannot describe this. Crystal plasticity (CP) is the constitutive model that can.

The figure pulls a model crystal that can slip on two families of planes. At φ = 45° one family lies at 45° to the load, and the crystal shears in blocks like a deck of cards. It yields at 2 τc, where τc is the shear stress that starts slip. Turn it to φ = 15° and it needs twice the stress. At φ = 165° the other family takes over, and the crystal is soft again. The dashed line is an isotropic model fitted to many such crystals: it gives one yield stress for every orientation, and it is wrong for each of them. Pull the crystal from zero and press ▶ to watch the blocks slide. The note explains this for one crystal in five steps. Step 6 puts many crystals together into a polycrystal, where the grains must share the load.

The problem

An isotropic model such as von Mises plasticity uses one yield surface for every direction. That works for a large piece of metal with many randomly oriented grains. It fails for a single crystal and for a part only a few grains across. It also fails for a sheet with a strong texture, a preferred orientation of its grains. It also cannot say how rolling or forging changes the texture, although the texture sets the anisotropy of the finished part (Dieter 1988).

Crystal plasticity models the mechanism instead: plastic flow is dislocation glide on discrete crystallographic slip systems. The key idea in one sentence: each slip system shears on its own, driven only by the shear stress resolved onto it, and the lattice carries the stress. CP is discrete dislocation dynamics averaged over a volume, with no transport of dislocations between material points. Crystal plasticity as averaged dislocation dynamics derives the three substitutions that perform this average. The price is cost: many slip systems, a hardening law for each, and a stiff local solve at every integration point.

History

Until the 1920s, engineers described yield with isotropic criteria. Under the microscope, a deformed crystal showed fine parallel lines on its surface: steps where blocks of the crystal had slid along one family of atomic planes. The theory then grew in two waves. Single-crystal physics came in the 1920s and 1930s, with small-strain, rate-independent theories solved by hand. The finite-strain, computational framework came from the 1970s on.

Step 1: Schmid’s law

A slip system $\alpha$ is a pair of unit vectors in the reference lattice: the slip plane normal $\mathbf m^\alpha$ and the slip direction $\mathbf s^\alpha$, with $\mathbf s^\alpha\cdot\mathbf m^\alpha=0$. The crystal’s Bravais lattice fixes which systems exist. Slip prefers the densest planes and the shortest lattice translation, because the shortest translation is the Burgers vector with the lowest energy:

• Face-centred cubic (FCC) metals (Cu, Al, Ni) have 12 systems of the $\{111\}\langle 110\rangle$ family: close-packed planes and close-packed directions.
• Body-centred cubic (BCC) metals (ferritic steel) have no truly close-packed plane. They slip along $\langle 111\rangle$ on $\{110\}$, $\{112\}$ and $\{123\}$ planes, 48 potential systems, with a strong temperature dependence.
• Hexagonal close-packed (HCP) metals (Ti, Mg, Zr) have a few easy basal and prismatic systems and hard pyramidal $\langle c+a\rangle$ systems. They are strongly anisotropic and often need twinning as well.

The driving force on a system is the resolved shear stress $\tau^\alpha$, the projection of the Cauchy stress onto the slip system. At small strain, $$\tau^\alpha = \boldsymbol\sigma : (\mathbf s^\alpha\otimes\mathbf m^\alpha) = \boldsymbol\sigma : \mathbf P^\alpha , \qquad \mathbf P^\alpha = \tfrac12(\mathbf s^\alpha\otimes\mathbf m^\alpha + \mathbf m^\alpha\otimes\mathbf s^\alpha) .$$ Step 2 gives the finite-strain form. For uniaxial tension $\sigma$ along a unit vector $\mathbf t$, this reduces to $\tau^\alpha = \sigma\cos\lambda\cos\chi$, with $\lambda$ and $\chi$ the angles from $\mathbf t$ to $\mathbf s^\alpha$ and $\mathbf m^\alpha$. The product $\cos\lambda\cos\chi$ is the Schmid factor $m$, at most 0.5. Schmid’s law says that a system starts to slip once $|\tau^\alpha|$ reaches the critical resolved shear stress $\tau_c$, also written $g^\alpha$. It replaces the single yield surface by one inequality per slip system.

The planar crystal of the figure has two systems 60° apart. At φ = 60° the two systems lie symmetric about the load, so both feel the same stress. At φ = 90° system 2 is active, and its arrow has turned round because its resolved shear stress has changed sign. At φ = 15° both Schmid factors are 0.25, the worst orientation. The yield stress therefore ranges from 2 τc to 4 τc. This range returns in Step 6 as the spread of yield stresses over the grains of a polycrystal.

Step 2: kinematics and the stress that drives slip

Crystal plasticity is most natural at finite strain, with the multiplicative split of the deformation gradient (Hill and Rice 1972): $$\mathbf F = \mathbf F^{\mathrm e}\,\mathbf F^{\mathrm p} .$$

• $\mathbf F^{\mathrm p}$ maps the reference body to a fictitious intermediate configuration. It shears material through the lattice, and leaves the lattice itself undistorted and unrotated.
• $\mathbf F^{\mathrm e}$ then stretches and rotates the lattice elastically, to the current configuration. Only $\mathbf F^{\mathrm e}$ causes stress.

In the figure the middle block is sheared, yet its lattice is the reference lattice. Only the right panel turns the blue lattice direction. A CP model therefore carries the lattice orientation, and so the texture, in $\mathbf F^{\mathrm e}$. The plastic velocity gradient is the sum of the shears on all systems: $$\mathbf L^{\mathrm p} = \dot{\mathbf F}^{\mathrm p}\mathbf F^{\mathrm p\,-1} = \sum_{\alpha} \dot\gamma^{\alpha}\,(\mathbf s^\alpha\otimes\mathbf m^\alpha) ,$$ where $\dot\gamma^\alpha$ is the shear rate on system $\alpha$. The shear rates are the central unknowns of the model.

The stress comes from the elastic part alone. The elastic strain is $\mathbf E^{\mathrm e} = \tfrac12(\mathbf C^{\mathrm e}-\mathbf I)$, with $\mathbf C^{\mathrm e} = \mathbf F^{\mathrm e\,T}\mathbf F^{\mathrm e}$. The second Piola–Kirchhoff stress in the intermediate configuration is $\mathbf S = \mathbb C:\mathbf E^{\mathrm e}$, where $\mathbb C$ is the stiffness of the lattice (Roters et al. 2010). Cubic crystals need three constants, $C_{11}$, $C_{12}$ and $C_{44}$. Most cubic metals are elastically anisotropic: copper is almost three times stiffer along a cube diagonal $\langle 111\rangle$ than along a cube edge $\langle 100\rangle$ (Dieter 1988).

Which stress drives slip follows from the stress power. It splits as $$J\,\boldsymbol\sigma:\mathbf L = \mathbf S:\dot{\mathbf E}^{\mathrm e} + \mathbf M:\mathbf L^{\mathrm p}, \qquad \mathbf M = \mathbf C^{\mathrm e}\mathbf S ,$$ with $J=\det\mathbf F$. The first term is stored elastically. The second is the plastic power, and $\mathbf M$ is the Mandel stress. Inserting $\mathbf L^{\mathrm p}$ gives $\mathbf M:\mathbf L^{\mathrm p} = \sum_\alpha \tau^\alpha\dot\gamma^\alpha$, so the resolved shear stress that does work on slip is $$\tau^\alpha = \mathbf M : (\mathbf s^\alpha\otimes\mathbf m^\alpha) .$$ Metals reach elastic strains of only about 0.1–1 %. Then $\mathbf M \approx \mathbf R^{\mathrm e\,T}\boldsymbol\sigma\,\mathbf R^{\mathrm e}$, with $\mathbf R^{\mathrm e}$ the lattice rotation, and Step 1’s formula holds with the slip vectors turned by $\mathbf R^{\mathrm e}$. The small-strain shortcut is accurate for the stress, but the rotation can still be large (Step 5).

Step 3: the flow rule

Underneath every flow rule sits Orowan’s equation. The shear rate on a system is the mobile dislocation density times the Burgers vector times the mean glide velocity, $\dot\gamma^\alpha = \rho_{\mathrm m}^\alpha b\, v^\alpha$. Flow rules differ only in how they supply the product. Phenomenological models fit it as a function of stress. Physics-based models keep $\rho_{\mathrm m}^\alpha$ explicit and prescribe a mobility, for example thermally activated glide.

• The rate-independent form takes $\dot\gamma^\alpha$ from Schmid’s law plus a consistency condition. It has a selection problem: when more than five systems reach $\tau_c$ together, the set of active systems is not unique.
• The rate-dependent (viscoplastic) power law removes the selection problem by letting every system slip a little (Hutchinson 1976; Peirce, Asaro, and Needleman 1983): $$\dot\gamma^\alpha = \dot\gamma_0 \left|\frac{\tau^\alpha}{g^\alpha}\right|^{1/m}\!\operatorname{sign}(\tau^\alpha) .$$ Here $\dot\gamma_0$ is a reference shear rate and $m$ the rate sensitivity exponent. The limit $m\to 0$ recovers the rate-independent law.

The figure shows what $m$ does. A 10 % overload multiplies the shear rate by 1.6 at $m = 0.2$ and by about 14,000 at $m = 0.01$. Metals at room temperature have $m$ between 0.005 and 0.05, close to the sharp threshold. The steep slope $1/m$ makes the local update stiff: explicit time integration needs a number of steps that grows like $1/m$. Solving crystal plasticity derives this bound and measures it.

The power law also satisfies the second law system by system. The plastic dissipation is $\sum_\alpha\tau^\alpha\dot\gamma^\alpha$, and each $\dot\gamma^\alpha$ has the sign of $\tau^\alpha$, so no term is negative. The law is the derivative of a dissipation potential, $$\dot\gamma^\alpha = \frac{\partial\Omega}{\partial\tau^\alpha}, \qquad \Omega = \sum_\alpha \frac{m\,\dot\gamma_0\,g^\alpha}{1+m}\left|\frac{\tau^\alpha}{g^\alpha}\right|^{(1+m)/m} .$$ This normality is Rice’s result for slip as an internal variable (Rice 1971). It links crystal plasticity to the generalized standard material framework.

Step 4: hardening

A face-centred cubic single crystal, pulled in single slip, hardens in three stages (Dieter 1988; Mecking and Kocks 1981):

• Stage I, easy glide. One system slips. Its dislocations meet few obstacles, and the hardening rate is low.
• Stage II. A second system starts. Dislocations of the two systems cut and block each other. The hardening rate jumps to a high, nearly constant value, a few thousandths of the shear modulus.
• Stage III. The hardening rate falls, because screw dislocations cross-slip around obstacles and annihilate. This is dynamic recovery.

Stage II shows that slip on one system hardens the others. A hardening matrix $h^{\alpha\beta}$ couples the systems. Its diagonal is self hardening and its off-diagonal is latent hardening: $$\dot g^\alpha = \sum_\beta h^{\alpha\beta}\,|\dot\gamma^\beta|, \qquad h^{\alpha\beta} = h\,\big[q + (1-q)\,\delta^{\alpha\beta}\big] .$$ Latent-hardening tests prestrain a crystal on one system and then load another. A system that has not slipped turns out harder than the one that has, with $q \approx 1$–$1.4$ for FCC metals (Asaro 1983). The two families of hardening law differ in what $h$ depends on:

• Phenomenological laws prescribe $h$ as a function of the total slip $\gamma_a = \int\sum_\alpha|\dot\gamma^\alpha|\,\mathrm dt$. The Peirce–Asaro–Needleman law, $h = h_0\,\mathrm{sech}^2\!\big(h_0\gamma_a/(\tau_s-\tau_0)\big)$, raises the slip resistance from $\tau_0$ to a saturation value $\tau_s$ (Peirce, Asaro, and Needleman 1983). The Voce law saturates exponentially in the same way.
• Dislocation-density laws carry a density $\rho^\alpha$ per system. The Taylor relation gives the resistance, $g^\alpha = \alpha_{\mathrm T}\,\mu b\sqrt{\textstyle\sum_\beta a^{\alpha\beta}\rho^\beta}$, where the interaction matrix $a^{\alpha\beta}$, averaged junction statistics, plays the part of $q$ (Roters et al. 2010). The Kocks–Mecking law evolves the density, $\mathrm d\rho/\mathrm d\gamma = k_1\sqrt{\rho} - k_2\,\rho$ (Mecking and Kocks 1981). The storage term $k_1\sqrt\rho$ gives the constant rate of stage II, and the recovery term $k_2\rho$ gives stage III and saturation.
• Back stress (kinematic hardening) shifts the threshold instead of raising it: the flow rule uses $\tau^\alpha - \chi^\alpha$. The Armstrong–Frederick law, $\dot\chi^\alpha = c\,\dot\gamma^\alpha - d\,\chi^\alpha|\dot\gamma^\alpha|$, gives the Bauschinger effect and ratcheting under reversed load (Frederick and Armstrong 2007). Any cyclic or fatigue use of crystal plasticity needs it.

The figure pulls a crystal oriented for single slip on system 1 (Schmid factors 0.49 and 0.32), with linear hardening. At q = 0, slip on system 1 does not harden system 2, which starts at 9 % strain. At q = 1, Taylor’s isotropic hardening, system 2 waits until 25 %. At q = 1.4 it waits until 47 %. In both cases the second system starts because the lattice rotates, which is Step 5. The model has linear hardening and no recovery, so it shows when the second system starts but not the change of hardening rate between stages.

Step 5: the lattice rotates

The spin of a material element splits into a lattice part and a slip part. The lattice spins at $$\mathbf W^{\mathrm e} = \mathbf W - \mathbf W^{\mathrm p}, \qquad \mathbf W^{\mathrm p} = \sum_\alpha \dot\gamma^\alpha\,\operatorname{skew}(\bar{\mathbf s}^\alpha\otimes\bar{\mathbf m}^\alpha),$$ where $\mathbf W$ is the total spin, $\mathbf W^{\mathrm p}$ the plastic spin, and $\bar{\mathbf s}^\alpha$, $\bar{\mathbf m}^\alpha$ the slip vectors in the current lattice. The boundary conditions set $\mathbf W$. Whatever rotation slip does not supply, the lattice must. Texture is this lattice spin, integrated over every grain.

For single slip in a tensile test the rule reduces to geometry (Schmid and Boas 1935):

• The grips hold the specimen axis fixed, so $\mathbf W$ cancels the rotation that slip would cause. Let $\lambda$ be the angle between the slip direction and the tensile axis.
• A material line along the axis has length $l$. Slip along $\mathbf s$ does not change its component normal to $\mathbf s$, so $l\sin\lambda = l_0\sin\lambda_0$, or $$\sin\lambda = \frac{\sin\lambda_0}{1+\varepsilon} .$$
• The slip direction therefore turns towards the tensile axis. The Schmid factor $\sin\lambda\cos\lambda$ changes with it, so the crystal hardens or softens by geometry alone.

In the planar figure the axis is vertical. The grips then give the lattice spin $\omega = \sum_\alpha\dot\gamma^\alpha\cos^2 b^\alpha$, with $b^\alpha$ the angle of the slip direction from the horizontal. At ε = 0 system 1 has the larger Schmid factor and slips alone. At ε = 35 % the lattice has turned to 60°, where both systems have the Schmid factor 0.43. Both slip, their rotations cancel, and the orientation is stable. With q = 1.4, latent hardening keeps system 2 inactive past the symmetric orientation. The crystal overshoots by 4.6° before system 2 starts, as real FCC crystals do by a few degrees (Asaro 1983). In a polycrystal every grain turns in its own way, and predicting the resulting texture was the first big success of CPFEM (Kalidindi, Bronkhorst, and Anand 1992).

Step 6: from one crystal to a polycrystal

A metal part is a polycrystal: many grains, each a crystal with its own orientation, joined at grain boundaries. Each grain obeys Steps 1 to 5, but the single crystal does not say how the grains share the load. A grain cannot deform freely, because its neighbours must still fit around it. It also cannot carry any stress it likes, because the forces across each boundary must balance.

The figure shows the two simplest answers. Under equal stress, at σ = 2.1 τc only the best-oriented grains slip. At σ = 3 τc most grains slip, each on its own system, and the red boundaries mark neighbours that no longer fit. Under equal strain, every grain deforms alike on two systems, so the grains fit. Their stresses now range from 2.2 to 4.7 τc, and the red boundaries mark forces that do not balance. The true response lies between the two curves.

• Equal stress (Sachs) keeps equilibrium between the grains and breaks compatibility. It gives a lower bound on the flow stress.
• Equal strain keeps compatibility and breaks equilibrium (Taylor 1938). It gives an upper bound. Every grain then needs enough independent slip systems for an arbitrary shape change: five in 3D, two in the planar model. Bishop and Hill chose the active systems by the principle of maximum work (Bishop and Hill 1951).
• The Taylor factor $M = \sum_\alpha|\dot\gamma^\alpha|/\dot\varepsilon$ is the total slip a grain needs per unit axial strain. Equal plastic work, $\sigma\dot\varepsilon = \tau_c\sum_\alpha|\dot\gamma^\alpha|$, gives the polycrystal yield stress $\sigma_y = \bar M\,\tau_c$.
  • Random FCC polycrystals have $\bar M \approx 3.06$ under equal strain, against about 2.2 under equal stress (Kocks 1970).
  • In the planar model, $M = \tfrac{4}{\sqrt3}\big(|\cos 2b| + |\cos(2b+120°)|\big)$, which averages to $16/(\pi\sqrt3) \approx 2.94$ over all orientations. The 46 grains of the figure give 2.98.
  • Kocks set both bounds against single-crystal data (Kocks 1970).
• Self-consistent models treat each grain as an ellipsoidal inclusion in the still-unknown average medium (Kröner 1961; Hill 1965). Berveiller and Zaoui gave a simple explicit form for the plastic accommodation (Berveiller and Zaoui 1978). Lebensohn and Tomé made it anisotropic and viscoplastic (VPSC), for the texture of low-symmetry metals (Lebensohn and Tomé 1993). These are mean-field models: one answer per orientation, with no actual neighbours (Mean-field homogenisation).
• Full-field models mesh every grain and solve equilibrium and compatibility together (Barbe et al. 2001). Cailletaud et al. pulled 200 grains of the nickel alloy IN600 under four boundary conditions (Cailletaud et al. 2003) 📄. Two lessons stand:
  • The boundary condition changed the average curve by about 6 %, so 200 grains are not yet a representative volume (Kanit et al. 2003).
  • The grain-to-grain stress spread was 2 to 2.6 times that of the Berveiller–Zaoui model, mostly in the hydrostatic pressure. A mean-field model with uniform elasticity cannot see that part.

What this left open: in classical crystal plasticity a grain boundary is only a change of orientation. Pile-ups, slip transmission and the grain-size effect need an internal length, which the theory does not have. The next section adds it.

The missing length: gradient crystal plasticity

Classical crystal plasticity has no internal length. A 10 µm grain and a 1 mm grain with the same orientations give the same response. A Hall–Petch grain-size effect must therefore be written into the hardening law by hand. Any softening, such as damage or cleavage, localises into one element. The ingredient that the local theory throws away is the gradient of plastic slip. That gradient lives where Step 6 left off: at grain boundaries and free surfaces.

What a slip gradient stores

• A slip gradient cannot happen in a perfect lattice. Where slip changes from point to point, the lattice must hold extra dislocations to stay continuous. These are the geometrically necessary dislocations (GNDs). The statistically stored dislocations (SSDs) of Step 4 are the rest, stored by random trapping.
• Nye’s dislocation density tensor $\boldsymbol\Lambda=\nabla\times\mathbf F^{\mathrm{p}}$ measures the GND content. In a crystal it splits into a GND density for each slip system, edge and screw separately. The isotropic theories of Gradient plasticity have only one effective plastic strain, so this split is what a crystal adds.
• The GND density brings in a length, and with it a size effect. Let slip fall from $\gamma$ in a grain’s middle to zero at its boundary, over the grain size $d$: $$\rho_{\mathrm{GND}} \sim \frac{|\nabla\gamma|}{b} \sim \frac{\gamma}{b\,d}, \qquad \tau_c \sim \mu b\sqrt{\rho_{\mathrm{GND}}} \sim \mu\sqrt{\frac{b\,\gamma}{d}} .$$ The flow stress then grows as $d^{-1/2}$, the Hall–Petch form, from geometry alone.
• The gradient of what matters too. A gradient of the plastic strain misses part of the incompatibility, because the full plastic distortion, including its plastic spin, carries the dislocation content. Gurtin’s later theory accounts for the Burgers vector and for the dissipation of plastic spin (GURTIN 2004) (abstract only, unread).
  • Distortion gradient plasticity is the branch built on this. In wire torsion it captures energetic strengthening with a quadratic defect energy, where strain-gradient theories need a non-quadratic one (Bardella and Panteghini 2015) (abstract only, unread).
  • A defect-energy nonlinearity added to fit torsion data may have been compensating for a missing plastic spin.

Three places a GND density can enter

The models differ mainly in where the GND density enters the slip law. Each choice predicts a different size effect.

• The first place is the slip resistance, where GNDs act as obstacles: $\tau_c=\tau_{c0}+\alpha_{\mathrm T}\mu b\sqrt{\rho_{\mathrm{SSD}}+\rho_{\mathrm{GND}}}$. This is Taylor hardening, and it is isotropic. The return mapping stays local, because the GND density is only read from the current slip field.
  • Busso et al. used it for two-phase γ/γ′ nickel single crystals, and passed the result to the component scale through a size-dependent hardening term (Busso 2000) 📄.
  • Haouala et al. computed the curl of $\mathbf F^{\mathrm{p}}$ in Fourier space and reached 100–200 grains with no fitted parameter (Haouala et al. 2020) 📄.
  • Keshavarz and Ghosh found grain-boundary hardening by inverting the Nye tensor, inside a three-scale model of nickel superalloys (Keshavarz and Ghosh 2015) 📄. Its polycrystal result uses one uniform sub-grain morphology and no realisation statistics.
  • The price: obstacles harden only after a gradient has formed, so initial yield does not depend on grain size. Every grain boundary is also opaque, with no boundary condition to change that.
  • Much of the vault’s crystal-plasticity fatigue literature uses exactly this term, so it runs lower-order gradient plasticity without calling it that.
• The second place is the resolved shear stress, from which a back stress is subtracted. The gradient of the GND density pushes back on further slip, so this is kinematic hardening. It gives a Bauschinger effect near grain boundaries.
• The third place is a higher-order stress from a gradient energy, with a balance law of its own. This is the full theory: it needs boundary conditions on slip at every grain boundary and free surface (Gurtin 2002) (not read; known through Evers et al. and Auth et al., who both cite it).

Evers, Brekelmans and Geers combine the first two in finite elements (Evers, Brekelmans, and Geers 2004) 📄. Their 18 GND densities are extra nodal fields. All densities act as Taylor obstacles, and the GND gradients add a back stress. The model gives a size effect on flow stress, and on initial yield when grain-boundary dislocations are present. The 12-grain test still gives a Hall–Petch exponent of 1.5, against 0.3–1.0 in experiments. The back stress also adds a second length besides $b$, its integration radius $R$ (3.16 µm).

Boundary conditions at a grain boundary

• A higher-order theory must say what slip does at a boundary. Two limits bracket the choice:
  • micro-hard: no plastic slip at the boundary, so dislocations pile up against it;
  • micro-free: no higher-order traction, so slip passes freely.
• Real boundaries lie between the limits, and the misorientation should decide where. Evers et al. block slip normal to the boundary and seed an initial GND density from the lattice mismatch. They are the only model here that derives the condition from the misorientation.
• The boundary condition is not a detail. In a strain-gradient unit cell of a nickel superalloy, the boundary conditions alone can halve the critical stress for flow softening (Choi et al. 2005) 📄. Softening there also needs a narrow matrix channel and a low-hardening matrix.

The cost, and how to keep the return mapping

• A higher-order theory adds global fields on every slip system. Evers et al. need 20 unknowns per node in 2D. The local return mapping, which every crystal plasticity code is built around, is also lost.
• The micromorphic approach moves the gradient onto a separate micro-variable, tied to $\mathbf F^{\mathrm{p}}$ by a penalty. The higher-order stress then enters the Schmid law as a back stress, and the local integration stays classical (Aslan et al. 2011) 📄.
  • The size-effect exponent ranges from 0 to −2, set by the penalty modulus, so Hall–Petch is included.
  • A single scalar micro-variable is the cheap version. It regularises slip bands and cleavage damage in real microstructures (Forest et al. 2014) 📄 (Lindroos et al. 2022) 📄.
  • A large penalty modulus makes the local Jacobian ill-conditioned. A strict Lagrange-multiplier formulation ran 30–200 times faster (Scherer et al. 2020).
• The obstacle route needs no extra global field at all. That is why it scales to hundreds of grains, and why it gives up the boundary conditions.

When the gradient meets a crack

• Auth, Brouzoulis and Ekh add phase-field fracture to a model whose gradient acts on a hardening variable per slip system (Auth, Brouzoulis, and Ekh 2024) 📄 (Ekh et al. 2007). The gradient energy acts along the slip direction only. It raises the yield threshold and is not a back stress.
• The boundary condition decides where the crack can start. A micro-hard specimen face removes damage from the side faces, and a micro-free face leaves the crack pattern almost unchanged.
• Damage irreversibility uses a micromorphic local phase field instead of a history variable. The study uses one single crystal, so how the gradient length $l_g$ relates to grain size is untested.

The models side by side

• The table shows two gaps. No two models share a benchmark, so obstacle, back stress and threshold have never been compared on the same data. Grain-boundary conditions are still mostly binary, and only Evers et al. derive one from the misorientation.
• The isotropic background, the energetic-against-dissipative controversy and the crack-tip applications stay in Gradient plasticity.

Where it is used

• Polycrystal homogenisation averages the single-crystal response over a representative volume of many grains, to get the anisotropic response of the material. The volume is normally built synthetically, with DREAM.3D or Neper, to match the measured grain shapes and orientation distributions of a real microstructure (Microstructure generation).
• Forming and texture: rolling and drawing textures, earing of deep-drawn cups, and forming limits.
• Fatigue crack initiation: slip localisation is crystallographic, so crystal plasticity gives grain-level precursors that $J_2$ plasticity cannot. Fatigue indicator parameter turns them into a life. Extreme-value statistics of fatigue indicator parameters turns them into a life distribution. Phase field crystal plasticity grows the crack itself.
• Computation: Solving crystal plasticity covers the implicit local update, the consistent tangent, finite element and spectral solvers, and DAMASK. Machine-learned replacements for the update are in Data-driven constitutive modelling.

Where it breaks

• Non-Schmid effects in BCC metals. Screw dislocations in BCC metals feel stress components other than the resolved shear, so Schmid’s law fails. The vault has no note on this yet.
• Twinning. HCP metals need it, and codes such as ρ-CP model it as pseudo-slip, but the vault has no note on it.
• Parameter identification. Fitting the flow and hardening parameters is non-unique. Cailletaud et al. call it the weakest link of the method (Cailletaud et al. 2003).
• Texture and grain boundaries have no notes of their own: no orientation distribution functions, pole figures or EBSD, and no boundary structure or slip transmission.
• Four debates are open:
  • What a free surface does. Electron-microscope studies report both harder and softer surface layers. Full-field crystal plasticity predicts a stress drop of about 10 % and fewer active systems at the surface (Cailletaud et al. 2003).
  • Mean-field or full-field. Full-field models give two to three times the grain-to-grain stress scatter. Which one predicts the tails that control fatigue initiation is still argued.
  • Where the slip gradient enters: obstacle, back stress or yield threshold. No two gradient models share a benchmark.
  • Grain-boundary conditions: micro-hard, micro-free, or derived from the misorientation. Only Evers et al. derive one.

Code

• crystal-plasticity-figure.html: the figure of this note. Its rot step integrates planar double slip with lattice rotation and latent hardening. Its demo step computes the equal-stress and equal-strain bounds and the Taylor factor of each grain.

Links:

• Dislocation theory — the map of the physical mechanism underneath.
• Crystal plasticity as averaged dislocation dynamics — the derivation of CP from discrete dislocation dynamics.
• Dislocation plasticity across scales — where CP sits on the ladder of models, and what it needs from the rungs below.
• Plasticity — the isotropic theory that CP refines; its polycrystal yield locus is the macroscopic shadow of Step 6.
• Generalized standard material — the dissipation-potential structure that the power law of Step 3 fits.
• Gradient plasticity — the isotropic background of the gradient section, with the energetic-against-dissipative controversy.
• Micromorphic approach — how CP gets an internal length while keeping the local return mapping.
• Solving crystal plasticity — the local and global solvers, and why the update must be implicit.
• Mean-field homogenisation — the self-consistent machinery behind the polycrystal models of Step 6.
• Homogenisation — representative volumes of many grains.
• Phase field crystal plasticity — crystal plasticity coupled to fracture, fatigue and microstructure evolution.

References: