Toward robust scalar-based gradient plasticity modeling and simulation at finite deformations

Abstract

Strain gradient plasticity has been the subject of extensive research in the past 40 years in order to model size effects in metal plasticity, on the one hand, and provide finite width shear bands in the simulation of localization phenomena, on the other hand. However, the use of the emerging models is still limited to academic applications and has not yet been adopted by industry practitioners. The present paper systematically reviews the pros and the cons of gradient plasticity at finite strains based on the gradient of scalar plastic variables, in particular the gradient of the cumulative plastic strain. It proposes benchmark tests addressing both size effect modeling and plastic strain localization simulation. It includes new analytical solutions for validation of FE implementation. It focuses on the micromorphic approach to gradient plasticity, as a convenient method for implementation in FE codes. New features of the analysis include the comparison of three distinct formulations of rate-independent gradient plasticity at finite deformations, based on the multiplicative decomposition of the deformation gradient and on quadratic potentials with respect to gradient terms. The performance of micromorphic and Lagrange-multiplier-based strain gradient plasticity models is evaluated for various monotonic and cyclic loading conditions including confined plasticity in simple glide and tension, bending and torsion at large deformations. Limitations are pointed out in the case of bending and torsion, which can be overcome for instance by the use of the gradient of equivalent plastic strain model.

Appendices

Appendix A

Consistent tangent matrices in the numerical implementation

The derivation of the consistent tangent matrix for a time-independent plastic model is detailed in this first Appendix. For the sake of brevity, only the model with a micromorphic variable associated with cumulative plastic strain is detailed in the following.

   The global resolution algorithm requires the following partial derivatives:

(74)

(75)

with

(76)

(77)

(78)

At Gauss point level, constitutive equations are integrated using a $\theta$-method [63]. The values of all integrated variables evaluated at an intermediate time designated by $\theta \in [0,1]$ are

$$\begin{aligned} {\mathcal {V}}_\textrm{int}^{t+\theta \varDelta t}={\mathcal {V}}_\textrm{int}^{t}+\theta \varDelta {\mathcal {V}}_\textrm{int}. \end{aligned}$$

(79)

The set of Eq. (57) can be gathered in the following form:

$$\begin{aligned} {\mathcal {R}}({\mathcal {V}}_\textrm{int}^{t+\theta \varDelta t},\varDelta {\mathcal {V}}_\textrm{int})=0. \end{aligned}$$

(80)

Since Eq. (80) is highly nonlinear, it is usually solved by means of a Newton method which requires the calculation of the Jacobian matrix

(81)

where is the normal to the yield surface. The value $\theta =1$ (implicit integration) is used in the present work for rate-independent plasticity.

Appendix B

Analytical solution for confined plasticity under shear

Consider the boundary value problem of Fig. 1a and introduced in Sect. 3.1. The strip is 2h-wide and infinite in the y-direction (invariant solution in this direction). In the case of a hardening plate (i.e., H $\ge$ 0), the microplastic variable $p_\chi$ is set to zero at $x=\pm h$ (Dirichlet higher-order boundary conditions). In the case of a softening plate (i.e., H < 0), $p_\chi$ is free at $x=\pm h$ far from the localization zone (Neumann higher-order conditions). The first balance equation reads

(82)

which yields

$$\begin{aligned} \sigma _{12,2}=0 ~ \textrm{and} ~ \sigma _{12,1}=0. \end{aligned}$$

(83)

Therefore, $\sigma _{12}=\tau$, taken positive without loss of generality, is uniform in the plate. The second balance equation is

(84)

where the constitutive equations for generalized stresses are given by

(85)

The differential equation governing the microplastic variable $p_\chi$ reads

$$\begin{aligned} \varDelta {p_\chi }-\dfrac{H_\chi }{A}(p_\chi -p)=0. \end{aligned}$$

(86)

The yield function is given by

(87)

where $\sigma _\textrm{eq}=\sqrt{3}~\tau$ is the von Mises stress. By combining Eqs. (86) and (87), the following partial differential equation for $p_\chi$ is obtained:

$$\begin{aligned} \varDelta {p_\chi }-\dfrac{HH_\chi }{A(H+H_\chi )}p_\chi +\dfrac{H_\chi }{A(H+H_\chi )}(\sigma _\textrm{eq}-R_0)=0. \end{aligned}$$

(88)

Three different cases can be distinguished: perfect plasticity ($H=0$), hardening ($H>0$), and softening ($H<0$) behavior.

1.1 B.1 Case 1: perfect plasticity ($H = 0$)

In this case, Eq. (88) reduces to

$$\begin{aligned} \varDelta {p_\chi }+\dfrac{1}{A}(\sigma _\textrm{eq}-R_0)=0 \end{aligned}$$

(89)

whose solution is

$$\begin{aligned} p_\chi (x)&=-\dfrac{\sqrt{3}\tau -R_0}{2A}x^2+C_1 x+C_2, \end{aligned}$$

(90)

where $C_1$ and $C_2$ are integration constants to be determined from boundary conditions:

$$\begin{aligned} p_\chi (x=\pm h)= 0 \implies C_1=0 \quad \text {and} \quad C_2=\dfrac{\sqrt{3}\tau -R_0}{2A}h^2. \end{aligned}$$

(91)

Finally, the fields of micromorphic deformation and cumulative plastic strain are

$$\begin{aligned} p_\chi (x)&=-\dfrac{\sqrt{3}\tau -R_0}{2A}(x^2-h^2),\end{aligned}$$

(92)

$$\begin{aligned} p(x)&=p_\chi (x)+\dfrac{\sqrt{3}\tau -R_0}{H_\chi }. \end{aligned}$$

(93)

Further, the expression of the uniform stress $\tau$ in the plate is

$$\begin{aligned} \tau =2\mu \varepsilon ^e_{12}=\dfrac{\mu }{h}\int _{-h}^{h}(\varepsilon _{12}-\varepsilon ^p_{12})\textrm{d}x =\mu {\overline{\gamma }}-\dfrac{\sqrt{3}\mu }{2h}\int _{-h}^{h}p(x) \textrm{d}x. \end{aligned}$$

(94)

Using Eq. (93), Eq. (94) reduces to

$$\begin{aligned} \tau =\dfrac{{\overline{\gamma }}+\dfrac{R_0}{\sqrt{3}}\left( \dfrac{3}{H_\chi }+\dfrac{h^2}{A}\right) }{\dfrac{1}{\mu }+\dfrac{3}{H_\chi }+\dfrac{h^2}{A}}. \end{aligned}$$

(95)

The parabolic profiles p(x) et $p_\chi (x)$ are illustrated by Fig. 26 and used for the validation of the FE implementation of the model. It is apparent in Fig. 26a that the value $H_\chi =10^5$ MPa ensures a very small difference $|p-p_\chi |$. It follows that the presented solution is almost identical to the solution of the same problem using the Aifantis strain gradient plasticity model. Increasing the parameter A flattens the profiles indicating that plastic deformation is more difficult to develop and higher stresses are reached. The limit $H_\chi \rightarrow \infty$ in Eq. (95) provides the shear stress level for the Aifantis model:

$$\begin{aligned} \tau = \dfrac{\mu }{A+\mu h^2}\left( A {\overline{\gamma }} + R_0h^2/\sqrt{3}\right) . \end{aligned}$$

(96)

The limit $A\rightarrow \infty$ shows that deformation is then purely elastic: $\tau = \mu {\overline{\gamma }}$. The plastic strain gradient would be too high to develop. In contrast, setting $A=0$ provides the classical elastic-perfectly plastic solution. The previous formula also reveals the apparent hardening modulus depending on A and the width h.

Fig. 26

a Analytic vs. numerical fields of cumulative plastic strain p and micromorphic variable $p_\chi$ for confined simple glide; b distribution of cumulative plastic strain for different values of the generalized modulus A. The profiles are given for the prescribed overall shear value ${\overline{\gamma }} =0.2$

1.2 B.2 Case 2: H > 0 (hardening)

For a strictly positive linear hardening modulus, the solution of Eq. (88) reads

$$\begin{aligned} p_\chi (x)=C_3\cosh (\omega x)+C_4 \sinh (\omega x)+\dfrac{\sigma _\textrm{eq}-R_0}{H}, \end{aligned}$$

(97)

where

$$\begin{aligned} \omega ^2=\dfrac{H H_\chi }{A(H+H_\chi )}. \end{aligned}$$

(98)

This formula defines the inverse characteristic length $\omega$ as a function of the plastic hardening modulus and higher-order parameters. The integration constants $C_3$ and $C_4$ are obtained by applying boundary conditions:

$$\begin{aligned} p_\chi (x=\pm h)= 0 \implies C_3= -\dfrac{\sigma _\textrm{eq}-R_0}{H \cosh (wh)} \text { and } C_4=0. \end{aligned}$$

(99)

It follows that

$$\begin{aligned} p_\chi (x)=\dfrac{\sqrt{3}\tau -R_0}{H}\left( 1-\dfrac{\cosh (\omega x)}{\cosh (\omega h)}\right) \end{aligned}$$

(100)

and

$$\begin{aligned} p(x)=p_\chi (x)+\dfrac{\sqrt{3}\tau -R_0}{H+H_\chi }\dfrac{\cosh (\omega x)}{\cosh (\omega h)}=\dfrac{\sqrt{3}\tau -R_0}{H}\left( 1-\dfrac{H_\chi }{H+H_\chi }\dfrac{\cosh (\omega x)}{\cosh (\omega h)}\right) . \end{aligned}$$

(101)

The value of $\tau$ is given by

$$\begin{aligned} \tau =\dfrac{{\overline{\gamma }}+R_0 Z_h }{\dfrac{1}{\mu }+\sqrt{3}Z_h}, \end{aligned}$$

(102)

where $Z_h=\dfrac{\sqrt{3}}{H}\left( 1-\sqrt{\dfrac{A H_\chi }{H(H+H_\chi )}}\dfrac{\tanh (\omega h)}{h} \right)$. These results are illustrated by Fig. 27. A clear difference $|p-p_\chi |$ is visible in Fig. 27a for a low value of the penalty modulus $H_\chi =10^3$ MPa. This difference almost vanished in Fig. 27b when $H_\chi$ is sufficiently high. This indicates again that the gradient plasticity model by [66] is a limit case of the micromorphic model as $H_\chi$ tends to infinity. The hyperbolic profiles can be recognized in Fig. 27c and d. Low values of the higher-order modulus A lead to flat distribution of plastic strain where high curvatures are reached for high values of A.

Fig. 27

The distribution of cumulative plastic strain for confined simple glide for various values of A and $H_\chi$. Material parameters: $R_0=20$ MPa, $H = 1500$ MPa. The profiles are given for the prescribed overall shear value ${\overline{\gamma }} =0.2$

1.3 B.3 Shear localization solution in micromorphic plasticity (Case 3: $H < 0$)

The development of a shear localization band in a homogeneous matrix strip is studied. The strip has a thickness of 2h in the x-direction and is infinite in the y-direction of the 2D shear plane. The stress state is homogeneous with

The localization band of finite width $2x_c<2h$ is entirely contained in the material strip. It is assumed that no plastic flow takes place outside the localization band so that the following zones can be defined:

• $-h \le x \le -x_c$: elastic domain, labeled with “−" superscript;

• $|x | \le x_c$: plastic domain, without any label;

• $x_c \le x \le h$: elastic domain, labeled with “+" superscript.

Periodicity boundary conditions are applied at the boundaries $x=\pm h$. Solutions are derived for the micromorphic plasticity model in the small deformation framework, using the standard von Mises plasticity yield function and softening modulus $H<0$. The limit case of the Aifantis strain gradient plasticity model is also obtained. The cumulative plastic strain field p(x) and the plastic microstrain $p_\chi (x)$ are functions of the sole variable x. The displacement field takes the following form:

$$\begin{aligned} u_x = {\overline{\gamma }} y, \quad u_y = u(x), \end{aligned}$$

(103)

where ${{\bar{\gamma }}}$ is the applied mean glide amount and u(x) is the unknown displacement fluctuation. The shear strain component is

$$\begin{aligned} \varepsilon _{12} = \frac{1}{2}({\overline{\gamma }}+ u') = \varepsilon _{12}^e +\varepsilon _{12}^p = \varepsilon _{12}^e + \frac{\sqrt{3}}{2} p, \end{aligned}$$

(104)

where $u'(x) = \textrm{d}u/\textrm{d}x$.

   The material is described by a linear hardening law with initial yield stress $R_0$ and negative hardening modulus $H< 0$.

1.3.1 B.3.1 Solution in the elastic domain

In the elastic domain, the micro-plastic strain $p^\pm _\chi (x)$ is the solution of the following differential equation:

$$\begin{aligned} p^\pm _\chi \,\!'' - \omega _\chi ^2 p^\pm _\chi = 0 \quad \textrm{with} \quad \omega _\chi = \sqrt{\frac{H_\chi }{A}}. \end{aligned}$$

(105)

This equation admits solutions of exponential type with wave number $\omega _\chi$. For symmetry reasons, assuming localization at the center of the plastic zone, the plastic microstrain and higher-order stress $b_x = A p_\chi '(x)$ are, respectively, even and odd functions. Since $b_x$ is periodic, it must vanish at the boundary (flat profile of microstrain):

$$\begin{aligned} p_\chi ^-\,\!'(-h) = p_\chi ^+\,\!'(h) = 0. \end{aligned}$$

(106)

It follows that

$$\begin{aligned} p_\chi ^-(x) = \alpha ^- \cosh (\omega _\chi (h+x)), \quad p_\chi ^+(x) = \alpha ^+ \cosh (\omega _\chi (h-x)), \end{aligned}$$

(107)

where $\alpha ^\pm$ are integration constants to be determined from boundary conditions.

1.3.2 B.3.2 Solution in the plastic domain

The yield conditions reads

$$\begin{aligned} \sqrt{3}\tau = R_0 + Hp - A p_\chi '' = R_0+ Hp_\chi - \frac{A(H+H_\chi )}{H_\chi } p_\chi ''. \end{aligned}$$

(108)

As a consequence of equilibrium, the shear stress $\tau$ is uniform. Due to the linear softening law $H<0$, the previous equation admits harmonic solutions with the wave number

$$\begin{aligned} \omega _{\chi p} = \sqrt{\frac{|H| H_\chi }{A(H + H_\chi )}}, \end{aligned}$$

(109)

assuming $H+H_\chi > 0$. This defines the inverse intrinsic length $\omega _{\chi p}$ in the plastic zone. The plastic microstrain profile takes the form

$$\begin{aligned} p_\chi (x) = \frac{\sqrt{3}\tau - R_0}{H} + C \cos (\omega _{\chi p} x). \end{aligned}$$

(110)

The plastic strain is obtained from the plastic microstrain field by the following equation, valid for $|x|\le x_c$:

$$\begin{aligned} p = p_\chi - \frac{A}{H_\chi } p_\chi '' = \frac{\sqrt{3}\tau - R_0}{H} + C\frac{H_\chi }{H+H_\chi } \cos (\omega _{\chi p} x) = \frac{\sqrt{3}\tau - R_0}{H} + C \left( 1 - \frac{\omega _{\chi p}^2}{\omega _\chi ^2}\right) \cos (\omega _{\chi p} x).\nonumber \\ \end{aligned}$$

(111)

The definition of the location $x_c$ is given by

$$\begin{aligned} p(\pm x_c)=0 = \frac{\sqrt{3}\tau - R_0}{H} + C\frac{H_\chi }{H+H_\chi } \cos (\omega _{\chi p} x_c). \end{aligned}$$

(112)

Four unknowns remain: $\alpha ^+, \alpha ^-, C, x_c$ to be determined from left-over continuity requirements.

1.3.3 B.3.3 Continuity conditions

The interface conditions to be enforced are the following:

• Continuity of microstrain at $x=x_c$: $p_\chi (x_c) = p_\chi ^+(x_c)$,

  $$\begin{aligned} \alpha ^+ \cosh (\omega _\chi (h-x_c)) = \frac{\sqrt{3}\tau - R_0}{H} + C \cos (\omega _{\chi p} x_c). \end{aligned}$$

  (113)

• Continuity of microstrain at $x=-x_c$: $p_\chi (x_c) = p_\chi ^-(-x_c)$

  $$\begin{aligned} \alpha ^- \cosh (\omega _\chi (h-x_c)) = \frac{\sqrt{3}\tau - R_0}{H} + C \cos (\omega _{\chi p} x_c). \end{aligned}$$

  (114)

  It follows from the two previous equations that

  $$\begin{aligned} \alpha ^+ = \alpha ^- = \alpha . \end{aligned}$$

  (115)

• Continuity of the higher-order stress component at $x_c$: $b_x(x_c) = b^+_x(x_c) \, \Longrightarrow \, p_\chi '(x_c) = p_\chi ^+\,\!'(x_c)$,

  $$\begin{aligned} \alpha ^+\omega _\chi \sinh (\omega _\chi (h-x_c)) = C \omega _{\chi p} \sin (\omega _{\chi p} x_c). \end{aligned}$$

  (116)

• Continuity of the higher-order stress component at $-x_c$: $b_x(-x_c) = b^-_x(-x_c) \, \Longrightarrow \, p_\chi '(-x_c) = p_\chi ^-\,\!'(-x_c)$. This condition turns out to be automatically fulfilled once the result (115) is taken into account.

Only three unknowns remain, namely $\alpha ,C,x_c$, which are determined from the three Eqs. (112), (113), and (116).

1.3.4 B.3.4 Transcendental equation for the plastic zone boundary

The equation to be solved for $x_c$ is obtained by computing the ratio of Eq. (116) by (113):

$$\begin{aligned} \tanh (\omega _\chi (h-x_c))= & {} \frac{\omega _{\chi p}}{\omega _\chi } \frac{C \sin (\omega _{\chi p} x_c)}{ \frac{\sqrt{3}\tau - R_0}{H} + C \cos (\omega _{\chi p} x_c)} \nonumber \\= & {} \frac{\omega _{\chi p}}{\omega _\chi } \frac{C \cos (\omega _{\chi p} x_c)}{ \frac{\sqrt{3}\tau - R_0}{H} + C \cos (\omega _{\chi p} x_c)} \tan (\omega _{\chi p}x_c) \nonumber \\= & {} \frac{\omega _{\chi p}}{\omega _\chi } \frac{H+H_\chi }{H} \tan (\omega _{\chi p}x_c) \end{aligned}$$

(117)

after elimination of $C \cos (\omega _{\chi p} x_c)$ term by means of (112).

   The location $x_c$ is therefore a zero of the function

$$\begin{aligned} f(y) = \tanh (\omega _\chi (h-y)) - \frac{\omega _{\chi p}}{\omega _\chi } \left( 1+\frac{H_\chi }{H}\right) \tan (\omega _{\chi p}y), \end{aligned}$$

(118)

which results in the announced transcendental equation.

   Once $x_c$ is determined, the constant C and $\alpha$ are computed from (112) and (116):

$$\begin{aligned} C = \frac{(R_0-\sqrt{3}\tau )(H+H_\chi )}{H H_\chi \cos (\omega _{\chi p}x_c)}, \quad \alpha = C\frac{\omega _{\chi p}\sin (\omega _{\chi p} x_c)}{\omega _\chi \sin (\omega _\chi (h-x_c))}. \end{aligned}$$

(119)

It remains to derive the relation between ${{\bar{\gamma }}}$ and $\tau$. This is done by means of the elasticity law:

$$\begin{aligned} \frac{\tau }{\mu } = {{\bar{\gamma }}} + u' - \sqrt{3}p. \end{aligned}$$

(120)

Integration of this equation over the interval $[-h,h]$, after accounting for the periodicity of u, provides the relation between shear stress and applied shear strain:

$$\begin{aligned} \frac{\tau }{\mu } = {{\bar{\gamma }}} -\frac{1}{h} \left( \frac{\sqrt{3}\tau -R_0}{H}x_c - \sqrt{3} h {\bar{p}}\right) , \end{aligned}$$

(121)

where the average plastic strain is

$$\begin{aligned} {\bar{p}}= & {} \frac{1}{2h}\int _{-x_c}^{x_c} p(x) \textrm{d}x \nonumber \\= & {} \frac{1}{h} \left( \frac{\sqrt{3}\tau -R_0}{H} x_c + C\left( 1-\frac{\omega _{\chi p}^2}{\omega _\chi ^2}\right) \sin (\omega _{\chi p} x_c)\right) \nonumber \\= & {} \frac{\sqrt{3}\tau -R_0}{hH} \left( x_c- \frac{\tan (\omega _{\chi p} x_c)}{\omega _{\chi p}}\right) . \end{aligned}$$

(122)

Finally,

$$\begin{aligned} {{\bar{\gamma }}} = \frac{\tau }{\mu } +\sqrt{3}\frac{\sqrt{3}\tau -R_0}{hH} \left( x_c - \frac{\tan (\omega _{\chi p} x_c)}{\omega _{\chi p}}\right) . \end{aligned}$$

(123)

The problem can therefore be solved for each given value of the shear stress $\tau$. The corresponding applied shear is computed from Eq. (123). Conversely, for prescribed shear ${{\bar{\gamma }}}$, the unknowns $\tau$ and $x_c$ are determined by solving the nonlinear system (118) and (123).

1.3.5 B.3.5 Limit case: strain gradient plasticity

The solution is straightforwardly found in the case of Aifantis strain gradient plasticity, either directly from the strain gradient plasticity equations or as a limit case of the previous micromorphic solution. The plastic field p(x) is the solution of the yield condition

$$\begin{aligned} \sqrt{3}\tau = R_0 + Hp - Ap'' \end{aligned}$$

(124)

in the whole plastic domain $|x|\le x_c$. The negative hardening modulus $H<0$ is responsible for the localization phenomenon. A harmonic solution with wave number

$$\begin{aligned} \omega _p = \sqrt{\frac{|H|}{A}} = \lim _{H_\chi \rightarrow \infty } \omega _{\chi p} \end{aligned}$$

(125)

is found. It is the limit of the micromorphic wave number (109) by increasing the penalty on the difference between the cumulative plastic strain p and the plastic microstrain $p_\chi$. The boundary of the plastic zone is defined by the condition

$$\begin{aligned} p(x_c) = 0 \quad \Longrightarrow \quad x_c = \frac{\pi }{\omega _p}. \end{aligned}$$

(126)

Finally, the localization band can be described by the following sinus branch:

$$\begin{aligned} p(x) = \frac{\sqrt{3}\tau - R_0}{H}(1+\cos (\omega _p x)), \end{aligned}$$

(127)

with maximum plastic strain $2(\sqrt{3}\tau - R_0)/H$ at $x=0$. Using the Hooke law (120) and periodicity of displacement, the relation between shear stress and shear strain is obtained:

$$\begin{aligned} \tau \left( \frac{1}{\mu }+ \frac{3}{H} \frac{x_c}{h}\right) = {{\bar{\gamma }}} + \frac{\sqrt{3}R_0}{H} \frac{x_c}{h}. \end{aligned}$$

(128)

This relation is also obtained from the micromorphic solution (123) in the limit $H_\chi \rightarrow \infty$, which leads to $\tan (\omega _{\chi p}x_c)\rightarrow 0$.

1.3.6 B.3.6 Example and discussion of multiple solutions

The previous solutions are illustrated in a specific case characterized by the parameters given in Table 5. The analytical solutions are compared to finite element simulations based on the micromorphic plasticity model at small deformations. The finite element simulation is illustrated by the deformed states of the strip and plastic microstrain fields of Fig. 28.

Table 5 Geometrical, loading, and material parameters for the simulation of shear localization in a strip

Fig. 28

Finite element simulation of shear localization in a micromorphic strip. Deformed states ${{\bar{\gamma }}} = 0.; 0.05; 0.1; 0.15; 0.2$, from top to bottom, respectively. The fields of plastic microstrain $p_\chi$ are also given. The parameters of the simulation are given in Table 5

   The limit case of strain gradient plasticity is illustrated by Fig. 29 where the analytical solution is compared to the FE simulations using the micromorphic model with the penalty parameter $H_\chi = 10^5$ MPa. The same excellent agreement is observed using the Lagrange-multiplier-based model.

   Figure 30 shows that the transcendental equation $f(x)=0$, see Eq. (118), admits three solutions for $x_c$ in the interval [0, h], namely $x_c \simeq 1.198, 2.604, 4.012$ mm.

   Figure 31 shows perfect agreement between the analytical and FE solutions, for the lowest value of $x_c$. The regularity of the $p_\chi (x)$ profile is clearly visible with vanishing tangents around $x=\pm 2$ mm. In contrast, the p(x) function is not differentiable at $\pm x_c$ and reaches higher peak value than the smoother microplastic strain $p_\chi (x)$.

   Figures 32 and 33 show the solutions obtained for the other possible values of $x_c$. They correspond to the existence of two or three coexisting bands. However, these solutions cannot be accepted because it is apparent that the cumulative plastic strain variable takes negative values at some places, which is forbidden. This means that these solutions must be reconsidered by taking possible elastic unloading into account. This explains why these two- or three-branch solutions are not found in the FE analysis. Note also that the number of finite width localization bands is limited by the size 2h of the strip element.

Fig. 29

Strain gradient plasticity solution of the shear localization problem with parameters listed in Table 5

Fig. 30

Roots of the transcendental equation $f(x)=0$ given by Eq. (118), with parameters listed in Table 5

Fig. 31

Comparison between analytical and FE solutions of the shear localization problem for the micromorphic plasticity model and the smallest positive solution of the transcendental equation, with parameters listed in Table 5

Fig. 32

Analytic solution of the shear localization problem using the second smallest positive value of $x_c$, with parameters listed in Table 5

Fig. 33

Analytic solution of the shear localization problem using the third smallest positive value of $x_c$, with parameters listed in Table 5

1.3.7 B.3.7 Regularity of the Laplacian term

It is instructive to analyze the profiles of the Laplacian of the plastic and microplastic variables since it plays a fundamental role in the modeling approach. In the strain gradient plasticity limit case, the function p(x) and its first derivative are continuous at $x=x_c$, as discussed in Sect. B.3.3 and illustrated in Fig. 29. In contrast, the second derivative

$$\begin{aligned} p''(x) = -\frac{\sqrt{3}\tau - R_0}{H}\cos (\omega _p x) \end{aligned}$$

(129)

is expected to experience a discontinuity at $x=x_c$. The discontinuity takes the value

$$\begin{aligned}{}[\![p''(x_c)]\!]= \frac{\sqrt{3}\tau - R_0}{H}\omega _p^2 \quad \textrm{since} \quad \cos (\omega _p x_c) = -1. \end{aligned}$$

(130)

With the parameters given in Table 5, the jump takes the value 1.47 mm$^{-2}$. This is clearly demonstrated by Fig. 34.

According to the micromorphic model, $p_\chi (x)$ and $p_\chi '(x)$ are continuous functions at $x=x_c$. The second derivative is computed as

$$\begin{aligned} p_\chi ''(x)= & {} -C\omega _{\chi p}^2 \cos (\omega _{\chi p}x) \quad (|x|< x_c), \\ p_\chi ^+\,\!''(x)= & {} \alpha \omega _\chi ^2 \cosh (\omega _\chi (h-x)) \quad (x_c<x<h), \\ [\![p_\chi ''(x_c)]\!]= & {} \alpha \omega _\chi ^2\cosh (\omega _\chi (h-x_c)) + C\cos (\omega _{\chi p} x_c) \omega _{\chi p}^2 = 0. \end{aligned}$$

The last result is obtained after consideration of Eqs. (113) and (119). The Laplacian of $p_\chi$ is therefore also continuous at $x_c$. This is a consequence of the second balance law (84) of the micromorphic model and constitutive laws for the generalized stresses a and ${{\varvec{\underline{b}}\,}}$. The latter equation results in PDE (86) which implies the continuity of the Laplacian $\varDelta p_\chi$ under the condition that p is continuous as it is the case in the present example. The Laplacian is therefore continuous as soon as $p-p_\chi$ is. This does not hold for the Laplacian of p(x). This is illustrated by Fig. 34 where the micromorphic response is compared to the constrained case. Note that $x_c=1.198$ mm in the micromorphic case, which is smaller than $x_c = 1.571$ mm found in the limit case of strain gradient plasticity. This ranking is also apparent in Fig. 34.

   Note that in the constrained strain gradient plasticity model the Lagrange multiplier $\lambda$ is directly proportional to the Laplace term. Its FE discretization with continuous shape functions is not compatible with the existence of discontinuities of the Laplacian. This may result in local oscillations around $x=x_c$ depending on the mesh size, see the discussion in [25].

1.3.8 B.3.8 Convergence of various energies to the strain gradient plasticity case

The free energy potential adopted in the considered example for the micromorphic model is

(131)

where $\underset{^\approx }{\varvec{C}}$ denotes the fourth-order tensor of elasticity. The micromorphic part of the free energy is

$$\begin{aligned} \rho \psi ^\chi (p,p_\chi ,\varvec{\nabla }p_\chi ) = \frac{1}{2} H_\chi (p-p_\chi )^2 + \frac{1}{2} A \varvec{\nabla }p_\chi \cdot \varvec{\nabla }p_\chi . \end{aligned}$$

(132)

In contrast, the Aifantis strain gradient plasticity model can be described by the following free energy potential:

(133)

The plastic strain gradient part of the free energy is

$$\begin{aligned} \rho \psi _\nabla (\varvec{\nabla }p) = \frac{1}{2} A \varvec{\nabla }p \cdot \varvec{\nabla }p. \end{aligned}$$

(134)

It is instructive to study the convergence of the micromorphic energy contribution $\psi ^\chi$ toward the gradient energy $\psi _\nabla$ in the limit $H_\chi \rightarrow \infty$, in the particular case of shear localization.

Fig. 34

Profiles of the Laplacian of plastic strain $p''(x)$ in the limit case of strain gradient plasticity (Aifantis model) and $p_\chi ''(x)$ in the micromorphic case, with parameters given in Table 5. Analytic and FE results are compared

Fig. 35

Various energy profiles in the shear localization zone. The predictions of the micromorphic model for two values of the penalty modulus ($H_\chi =100$ MPa on the left, $H_\chi =1000$ MPa on the right) are compared to the strain gradient plasticity solution. The other parameters are taken from Table 5

The profiles of the various contributions to the free energy of the micromorphic model are drawn in Fig. 35 for two values of the penalty modulus: $H_\chi =100$ MPa and $H_\chi =1000$ MPa. In the more constrained case ($H_\chi =1000$ MPa), the gradient energy $\psi _\nabla$ is found to almost coincide with the micromorphic energy $\psi ^\chi$, the penalty contribution $H_\chi (p-p_\chi )^2/2$ being negligible.

   For the lower value $H_\chi =100$ MPa, the energy densities $\psi _\nabla$ and $\psi ^\chi$ differ significantly. This is due, on the one hand, to the non–negligible contribution of the $H_\chi (p-p_\chi )^2/2$ term and, on the other hand, to high values of the gradient micromorphic contribution

$$\begin{aligned} \rho \psi ^\chi _{\nabla }(\varvec{\nabla }p_\chi ) = \frac{1}{2} \varvec{\nabla }p_\chi \cdot \varvec{\nabla }p_\chi \end{aligned}$$

compared with $\psi _\nabla$.