1 Introduction

In several plasticity theories, the assumption of rate-independence can be readily found as cornerstone of the formulation. Even though plasticity is generally a rate-dependent phenomenon, for some materials undergoing slow processes the material behavior can be assumed as rate-independent [12]. This assumption renders the transition between the elastic and elasto-plastic material behavior non-differentiable. Numerically, this introduces an additional computational cost as the yield function has to be evaluated at each quadrature point in order to determine whether yielding is occurring or not. In crystal plasticity the computational cost is increased as each slip system has its own yield function. This is further exacerbated in theories that involve gradients of the quantities capturing the inelastic behavior of the material in the yield function, such as the one implemented in this work. Furthermore, if the amount of active slip systems exceeds the degrees of freedom of the plastic strain tensor controlled by the flow rule, the consistency condition can not determine a unique set of plastic slip rates [4]. There are strategies to tackle the uniqueness issue within a rate-independent framework, e.g., one may choose the set of slip systems that minimize the energy dissipated during the process as proposed in Ref. [16]. Nonetheless, this method is computationally expensive and does not solve the non-uniqueness issue. Alternatively, oftentimes the rate-independent behavior is approximated by a rate-dependent formulation, in which the concept of elastic ranges is no longer valid and all slip systems are constantly active. As such, every kinematic process is intrinsically plastic. The latter approach can be readily found in the literature, e.g., Refs. [3, 14, 15] to name a few. Nonetheless, the fidelity of the rate-independent approximation has a numerical limit as the behavior it seeks to reproduce is non-differentiable; thus, the numerical solution of the non-linear problem becomes challenging as the approximation increases in accuracy. Therefore, the convergence of the non-linear solution scheme plays a deciding role in how well the rate-independent behavior is resolved. Other numerical difficulties resulting from complex constitutive models and critical values of material parameters may further influence to what extent the rate-independent limit can be achieved. As such, it becomes relevant to be aware of what non-desired effects are introduced in the material response and how negligible or dominant they actually are. In complex mathematical models these effects may be mistakenly perceived as intrinsic material behavior of the model and not as fictitious effects introduced by an insufficient rate-dependent approximation. Motivated by this particular scenario, here the problem of a infinite strip under shear loading is solved using a visco-plastic regularized gradient crystal plasticity formulation in order to identify and - for the specific boundary value problem solved - measure what we recognize as spurious effects in a rate-independent context resulting from the visco-plastic approximation.

In Sect. 2 the mathematical model to describe the gradient-enhanced elasto-plastic behavior in single crystals is presented. As the formulation is based on the assumption of rate-independence, the implemented viscoplastic regularization based on a set of sigmoid functions is showcased in Sect. 3. The boundary value problem of a single crystal subjected to simple shear is stated and numerically solved in Sect. 4. Finally, conclusions are drawn and summarized in Sect. 5.

Notation Zeroth, first and second order tensors are denoted by regular lowercase symbols, bold lowercase symbols and bold uppercase symbols, e.g. a, \( \varvec{a} \) and \( \varvec{A} \), respectively.

The Cartesian coordinate system is represented by the basis vectors \(\lbrace \varvec{e} _{\textrm{x}}, \varvec{e} _{\textrm{y}}, \varvec{e} _{\textrm{z}}\rbrace \). The ”\(\otimes \)” operator denotes the outer product, e.g. \( \varvec{a} \otimes \varvec{b} = a_i b_j \varvec{e} _i \otimes \varvec{e} _j\). Contractions are represented by the ”\(\varvec{\cdot }\)” operator, e.g. \( \varvec{A} \varvec{\cdot } \varvec{B} = A_{ij}B_{jk} \varvec{e} _i \otimes \varvec{e} _k\) and \( \varvec{A} \mathbin {{\varvec{\cdot }}{\varvec{\cdot }}} \varvec{B} = A_{ij}B_{ij}\). The cross product is denoted by the ”\(\times \)” operator, e.g. \( \varvec{a} \times \varvec{b} = \varepsilon _{ijk}a_j b_k \varvec{e} _i\), where \(\varepsilon _{ijk}\) are the components of the Levi-Civita tensor. The action of the Nabla operator \(\nabla = \partial \left( \bullet \right) / \partial x_i \varvec{e} _i\) on a tensor is determined by the basis vector on which it acts upon, e.g. \( \varvec{a} \oplus \nabla = \partial a_i / \partial x_j \varvec{e} _i \oplus \varvec{e} _j\) and \(\nabla \oplus \varvec{a} = \partial a_j / \partial x_i \varvec{e} _i \oplus \varvec{e} _j\) in Cartesian coordinates, where ”\(\oplus \)” can be any of the aforementioned operators.

2 Gradient crystal plasticity

Inelastic deformation in crystals is mainly attributed to the motion of line defects known as dislocations, specifically their gliding. The motion is quantified by a plastic slip \(\gamma _\alpha \), i.e., an inelastic shear deformation on a slip plane with unit normal \( \varvec{m} _\alpha \) along a slip direction \( \varvec{s} _\alpha \) in each slip system \(\alpha \in {{\mathbb {Z}}}^{+} \cap \left[ 1, n_{\textrm{s}}\right] \) with \(n_{\textrm{s}}\) being the total number of slip systems in the mono-crystalline structure. In classical single-crystal plasticity each slip system is endowed with its own yield function and flow rule. A gradient-enhanced extension of the classical theory was proposed by Gurtin [10], where a vectorial microforce \( \varvec{\xi } _\alpha \) and a scalar microforce \(\pi _\alpha \) are introduced as energetic conjugates of the slips \(\gamma _\alpha \) and their gradient, respectively. Following the principle of virtual power renders

(1)
(2)

where \( \varvec{T} \) is the symmetric stress tensor, \( \varvec{b} \) the body forces, \(\tau _\alpha = \varvec{T} \mathbin {{\varvec{\cdot }}{\varvec{\cdot }}}\left[ \varvec{s} _\alpha \otimes \varvec{m} _\alpha \right] \) the resolved shear stress, \( \varvec{x} \) the position vector and \({{{\mathcal {B}}}}\) the material domain. Equations 1 and 2 are known as the local quasi-static linear momentum balance and the local microforce balance, respectively. The latter is equivalent to a non-local yield condition for each slip system. The gradient enhancement introduces a length scale to the formulation; thus, allowing the description of size-dependent behavior. Furthermore, it offers a linkage between continuum mechanics and the geometrically necessary dislocations [12].

2.1 Kinematics

The infinitesimal theory assumes the additive decomposition of the displacement gradient into elastic and plastic parts, i.e.,

$$\begin{aligned} \varvec{u} \otimes \nabla = \varvec{H} _{\textrm{e}} + \varvec{H} _{\textrm{p}}. \end{aligned}$$
(3)

The linear strain measure allows per definition, \( \varvec{E} {:}{=}{{\,\textrm{sym}\,}}\left( \varvec{u} \right. \)\( \left. \otimes \nabla \right) \), the analogous decomposition. The plastic part of the deformation gradient is defined through the slip systems as

$$\begin{aligned} \varvec{H} _{\textrm{p}} {:}= \sum _\alpha \gamma _\alpha \varvec{s} _\alpha \otimes \varvec{m} _\alpha . \end{aligned}$$
(4)

As a measure of the geometric necessary dislocations the Burgers tensor [6] is given for the infinitesimal case [7] by

$$\begin{aligned} \varvec{G} {:}{=}\nabla \times \varvec{H} _{\textrm{p}}^{\textrm{T}}, \end{aligned}$$
(5)

which eventually admits the decomposition [9]

(6)

where and are, respectively, the edge and screw dislocation densities with the vector \( \varvec{l} _\alpha = \varvec{m} _\alpha \times \varvec{s} _\alpha \).

2.2 Consitutive equations

Following the Coleman-Noll procedure [8] in the energetic framework proposed by Gurtin [9], the ansatz for the free energy density

(7)

where the second term is the defect energy as proposed in Ref. [13]. This ansatz leads to the constitutive expressions for the stress and the vectorial microstress

(8)
(9)

where \( \overset{{\langle 4\rangle }}{\varvec{C}} \) is the elasticity tetrad (for the sake of simplicity considered isotropic in the following), \(\sigma _0\) the initial slip resistance and \(\ell \) an energetic length scale. In the following the defect energy takes on its quadratic form, i.e., \(n=2\), which leads to linear hardening during plastic flow. While it has been shown that non-quadratic defect energies, specifically \(n\in \left[ 1,2\right) \), are able to better replicate the results of discrete dislocation simulations, e.g., Ref. [13], they also lead to non-linear material behavior. Consequently, the choice of a quadratic defect energy is made in order to more easily identify the effects product of an insufficient visco-plastic regularization.

It is assumed that the dissipation density has the form \(\sum _\alpha \sigma _\alpha \left|{\dot{\gamma }}_\alpha \right|\) as done in Ref. [10]; thus, the scalar microstress is given by

$$\begin{aligned} \pi _\alpha = \sigma _\alpha {{\,\textrm{sgn}\,}}\left( {\dot{\gamma }}_\alpha \right) , \end{aligned}$$
(10)

where \(\sigma _\alpha \) is the slip resistance of the \(\alpha \)-th slip system. Its evolution equation is given by

$$\begin{aligned} {\dot{\sigma }}_\alpha = \sum _\beta \kappa \left[ q + \left[ 1-q\right] \delta _{\alpha \beta }\right] {{\,\textrm{sgn}\,}}\left( {\dot{\gamma }}_\beta \right) \end{aligned}$$
(11)

with the initial condition \(\sigma _\alpha = \sigma _0\) at \(t=0\). The material parameters in Eq. 11 are the linear hardening modulus \(\kappa \) and the latent to self hardening ratio q.

3 Viscoplastic regularization

In rate-independent plasticity elastic ranges are introduced, which are delimited by a yield surface inside which all kinematic processes are elastic. In the gradient crystal plasticity formulation described in Sect. 2, the elastic range of the \(\alpha \)-th slip system is given by

$$\begin{aligned} - \sigma _\alpha< \tau _\alpha + \varvec{\xi } _\alpha \varvec{\cdot } \nabla < \sigma _\alpha , \quad {\dot{\gamma }}_\alpha = 0. \end{aligned}$$
(12)

Once a plastic process starts, the slip system activates and the material flow is described by Eq. 2, i.e.,

$$\begin{aligned} \tau _\alpha + \varvec{\xi } _\alpha \varvec{\cdot } \nabla = \sigma _\alpha {{\,\textrm{sgn}\,}}\left( {\dot{\gamma }}_\alpha \right) , \quad {\dot{\gamma }}_\alpha \ne 0. \end{aligned}$$
(13)

The evaluation of the yield condition, Eq. 12, to determine which slip systems are active is not trivial, in particular since it involves second order derivatives of the plastic slips. A numerical alternative commonly implemented is a viscoplastic regularization. This is achieved in the presented formulation by an smooth approximation of the signum function, which can be accomplished by a sigmoid function

$$\begin{aligned} \pi _\alpha = \sigma _\alpha {{\,\textrm{sgn}\,}}\left( {\dot{\gamma }}_\alpha \right) \approx \sigma _\alpha {{\,\textrm{sig}\,}}_{\lambda }\left( {\dot{\gamma }}_\alpha \right) \end{aligned}$$
(14)

where \(\lambda \) is a regularization parameter controlling the accuracy of the approximation as its limit \(\lambda \rightarrow 0\) recovers the signum function. The regularization renders the theory rate-dependent and eliminates the aforementioned elastic ranges; thus, every process is intrinsically plastic, i.e., \({\dot{\gamma }}_\alpha \ne 0\).

Fig. 1
figure 1

(a) The sigmoid functions listed in Eqs. 15 to 19 and (b) their derivative using a regularization parameter of \(\lambda =1\)

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In the present work a selected set of sigmoid functions is investigated. Specifically,

$$\begin{aligned} {{\,\textrm{sig}\,}}_{\lambda }^{\textrm{at}}&= \frac{2}{\uppi } {{\,\textrm{atan}\,}}\left( \frac{\uppi }{2} \frac{x}{\lambda }\right) , \end{aligned}$$
(15)
$$\begin{aligned} {{\,\textrm{sig}\,}}_{\lambda }^{\textrm{sq}}&= \frac{x}{\sqrt{x^2 + \lambda ^2}}, \end{aligned}$$
(16)
$$\begin{aligned} {{\,\textrm{sig}\,}}_{\lambda }^{\textrm{gd}}&= \frac{2}{\uppi } {{\,\textrm{gd}\,}}\left( \frac{\uppi }{2}\frac{x}{\lambda }\right) , \end{aligned}$$
(17)
$$\begin{aligned} {{\,\textrm{sig}\,}}_{\lambda }^{\textrm{th}}&= \tanh \left( \frac{x}{\lambda }\right) , \end{aligned}$$
(18)
$$\begin{aligned} {{\,\textrm{sig}\,}}_{\lambda }^{\textrm{er}}&= {{\,\textrm{erf}\,}}\left( \frac{\sqrt{\uppi }}{2}\frac{x}{\lambda }\right) , \end{aligned}$$
(19)

where \({{\,\textrm{gd}\,}}\) and \({{\,\textrm{erf}\,}}\) are the Gudermannian and the error function, respectively. Equations 15 to 19 are graphically presented alongside their derivatives in Fig. 1. The functions are listed in ascending order with respect to their fidelity to the signum function with \({{\,\textrm{sig}\,}}_{\lambda }^{\textrm{at}}\) and \({{\,\textrm{sig}\,}}_{\lambda }^{\textrm{er}}\) being the worst and the best approximation of the chosen set, respectively.

4 Numerical investigation

The viscoplastic regularization approximates the rate-independent limit by introducing a certain amount of rate-dependence in the formulation, which is controlled in the present theory by the regularization parameter \(\lambda \). As an approximation, the material response deviates from the rate-independent case to some extent. In the following, a simple boundary value problem is numerically analyzed in order to make an assessment of these deviations and their relation to the regularization parameter \(\lambda \) and the sigmoid function used.

4.1 Boundary value problem

The problem of an infinite strip subjected to simple shear, as depicted in Fig. 2, is solved in two-dimensions under the assumption of plane strain. The boundary value problem is given by Eqs. 1 and 2 supplemented by the boundary conditions

$$\begin{aligned} \varvec{u}&= {h\varepsilon \, \varvec{e} _{\textrm{x}},}&\gamma _{\alpha }&= 0,&\forall \varvec{x}&\in {{\mathcal {S}}}_{\text {t}}, \end{aligned}$$
(20)
$$\begin{aligned} \varvec{u}&= \varvec{0} ,&\gamma _{\alpha }&= 0,&\forall \varvec{x}&\in {{\mathcal {S}}}_{\text {b}} \end{aligned}$$
(21)

and periodic boundary conditions on \({{\mathcal {S}}}_{\textrm{l}}\) and \({{\mathcal {S}}}_{\textrm{r}}\). The domain \({{{\mathcal {B}}}}\) is assumed to be composed of a single crystal with two slip systems defined by

$$\begin{aligned} \varvec{s} _\alpha&= {\cos \left( \theta _\alpha \right) { \varvec{e} _{\textrm{x}}} + \sin \left( \theta _\alpha \right) \varvec{e} _{\textrm{y}}} \end{aligned}$$
(22)
$$\begin{aligned} \varvec{m} _\alpha&= {-\sin \left( \theta _\alpha \right) \varvec{e} _{\textrm{x}} + \cos \left( \theta _\alpha \right) \varvec{e} _{\textrm{y}}} \end{aligned}$$
(23)

with \(\theta _\alpha = \lbrace -{60}{^{\circ }}, +{60}{^{\circ }}\rbrace \). The material parameters used in the sequel are shown in Table 1 and were taken from Ref. [14], in which the material parameters of the superalloy Inconel 718 were obtained through an inverse parameter identification by fixing some parameters, e.g., the Poisson ratio to \(\nu = {0.3858}\) (taken from Ref. [11]), while varying others until good accordance was obtained with experimental data. For a more detailed explanation refer to Ref. [14]. In order to study the effects of approximating the rate-independence via a viscoplastic regularization instead of recurring to a purely rate-independent formulation, different values of the regularization parameter \(\lambda \) are used. Furthermore, the sigmoid function used to approximate the signum function in Eq. 10 is also varied. It is worth noting that the material parameters taken from Ref. [14] are in principle only representative of Inconel 718 for \(\lambda = {3 \times 10^{-4}}\) as the regularization factor was fixed to this value during the inverse parameter identification.

Fig. 2
figure 2

Domain of the boundary value problem

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An analytical solution to the problem was formulated by Bittencourt et al. [5] for the case of \(q=1\) and a similar constitutive law for the vectorial microstress. They show that the shear component \(T_{12}\) of the stress tensor is constant over the strip’s height, whereas the plastic slips showcase a parabolic distribution along the y-direction. Due to the symmetry of the slip systems, the plastic slip fields are equal, i.e., \(\gamma _1 = \gamma _2\). These qualitative results also hold for the presented formulation and the case \(q=1.4\). As such, the sample point is chosen at \(y = {h}/{2}\) where the plastic slips reach their maximum value.

Table 1 Material parameters
Full size table

The load is divided into a sinusoidal loading and a cyclic phase, as shown in Fig. 3, with the transition between the loading phases occurring at \(t_{\textrm{t}} = 1\). The maximum and minimum values of the strain load are set to \(\epsilon _{\textrm{max}} = {0.02}\) and \(\epsilon _{\textrm{min}} = {0.01}\). The period of the cyclic load and the number of cycles are - unless otherwise specified - set to \(\tau = 1\) and \(n= 10\), respectively. A constant load step size of \(\Delta t = {5 \times 10^{-3}} \) is used throughout all simulations. The rates in the formulation, i.e., \({\dot{\gamma }}_\alpha \) and \({\dot{\sigma }}_\alpha \) are temporally discretized by the backward Euler method.

Fig. 3
figure 3

Evolution of the strain load on the upper boundary over time

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The continuum problem was numerically solved using the finite-element-method. The discrete domain is made up of a single column of 500 square finite elements, composed by the tensor product of a 2\(^{{\textrm{nd}}}\) order vector-valued and two 1\(^{{\textrm{st}}}\) order scalar-valued Lagrange elements for the displacement and slip fields, respectively.

The numerical implementation was realized based on the deal.II library [2]. The temporal evolution of the fields at the sample point presented in the next section were extracted from the raw results using Paraview [1] and post-processed in a Python [17] script.

4.2 Results

As stated in Sect. 3, the viscoplastic regularization eliminates the elastic ranges characteristic of rate-independent plasticity; thus, every process is inelastic and contributes to the further evolution of the slips. This can be verified by plotting one of the plastic slips over time as shown in Fig. 4 for \({{\,\textrm{sig}\,}}_{\lambda }^{\textrm{er}}\) over 100 cycles. In the rate-independent case, the plastic slips remain constant throughout the cyclic phase. As expected, this does not hold for the viscoplastic regularization. As the value of \(\lambda \) becomes higher the approximation becomes insufficient and significant plastic flow can be observed. The continuous evolution of the plastic slips reflects in all physical quantities that depend on the plastic slips, e.g., elastic strain and stress tensors. As a result, the rate-dependency may translate to behavior that can be mistaken as intrinsic to the material model, specially in complex theories and coupled problems. The simplicity of the boundary value problem allows to identify what in the following are designated as fictitious effects.

Fig. 4
figure 4

Evolution of the first plastic slip at \(y=h/2\) during the load shown in Fig. 3 with 100 cycles using \({{\,\textrm{sig}\,}}_\lambda ^{\textrm{er}}\) for different values of the regularization parameter

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Fig. 5
figure 5

Evolution of the shear component \(T_{12}\) of the stress tensor in the time interval \(\left[ 0,t_{\textrm{t}}\right] \) using \({{\,\textrm{sig}\,}}_{\lambda }^{\textrm{at}}\) with different values of the regularization parameter

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4.2.1 Fictitious softening

Table 2 Evaluation of \(m_{\textrm{s}}\) for different sigmoid functions and values of the regularization parameter
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Table 3 Evaluation of \(m_{\textrm{a}}\) for different sigmoid functions and values of the regularization parameter
Full size table

The first focus is to analyze the material response under static loading. For that purpose only the numerical results in the time interval \(\left[ 0, t_{\textrm{t}}\right] \) are considered. The evolution of the shear component \(T_{12}\) of the stress tensor is shown in Fig. 5 for different values of the regularization parameter. As anticipated, the transition from elastic to plastic behavior becomes more pronounced as the regularization parameter decreases. Unexpectedly, at the end of the loading phase a clear softening of the material can be observed for \(\lambda = {}{e-3}{}\) and \(\lambda = {}{e-4}{}\). The softening becomes smaller as the signum function is better approximated. In order to quantify this fictitious softening, the measure

$$\begin{aligned} m_{\textrm{s}} {:}{=} 1 - \frac{{\hat{T}}_{12} \left( t_{\textrm{t}}; {{\,\textrm{sig}\,}}_{\lambda }\right) }{\max \limits _{t \in \left[ 0,t_{\textrm{t}}\right] } \left( {\hat{T}}_{12} \left( t; {{\,\textrm{sig}\,}}_{\lambda }\right) \right) } \end{aligned}$$
(24)

is proposed. It compares the value of \(T_{12}\) at the transition time to its maximum value during the loading phase. Its values for different sigmoid functions and values of the regularization parameter are presented in Table 2. No fictitious softening occurs for \({{\,\textrm{sig}\,}}_{\lambda }^{\textrm{er}}\), \({{\,\textrm{sig}\,}}_{\lambda }^{\textrm{th}}\) and \({{\,\textrm{sig}\,}}_{\lambda }^{\textrm{gd}}\) starting from \(\lambda = {}{e-5}{}\). A reduction of two orders of magnitude is required by \({{\,\textrm{sig}\,}}_{\lambda }^\textrm{sq}\) to eliminate the fictitious behavior. Even for the smallest value of the regularization parameter \({{\,\textrm{sig}\,}}_{\lambda }^\textrm{at}\) still showcases some softening. Taking \({{\,\textrm{sig}\,}}_{{}{e-8}{}}^{\textrm{er}}\) as a control value, the accuracy measure

$$\begin{aligned} m_{\textrm{a}} {:}{=} 1 - \frac{{\hat{T}}_{12} \left( t_{\textrm{t}}; {{\,\textrm{sig}\,}}_{\lambda }\right) }{ {\hat{T}}_{12} \left( t_{\textrm{t}}; {{\,\textrm{sig}\,}}_{{}{e-8}{}}^{\textrm{er}}\right) } \end{aligned}$$
(25)

is defined. Its values are listed in Table 3. Analogous to the previous outcome, \({{\,\textrm{sig}\,}}_{\lambda }^{\textrm{er}}\), \({{\,\textrm{sig}\,}}_{\lambda }^{\textrm{th}}\) and \({{\,\textrm{sig}\,}}_{\lambda }^{\textrm{gd}}\) produce the best results by reaching the control value with \(\lambda = {}{e-7}{}\), which is one order of magnitude larger than the value required by \({{\,\textrm{sig}\,}}_{\lambda }^{\textrm{sq}}\) to reach the control value. In the case of \({{\,\textrm{sig}\,}}_{\lambda }^{\textrm{at}}\), the control value is not reached even with the smallest value of the regularization parameter.

4.2.2 Fictitious fatigue

Fig. 6
figure 6

Evolution of the shear component \(T_{12}\) of the stress tensor in the time interval \(\left[ 0, t_{\textrm{t}} + 10\tau \right] \) using \({{\,\textrm{sig}\,}}_{\lambda }^{\textrm{sq}}\) with different values of the regularization parameter

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The material behavior under cyclic loading is investigated by considering the results in the time interval \(\left[ t_{\textrm{t}}, t_{\textrm{t}} + n\tau \right] \) with n being the number of cycles. The evolution of the shear component \(T_{12}\) of the stress tensor over 10 cycles is shown in Fig. 6 for \({{\,\textrm{sig}\,}}_{\lambda }^{\textrm{sq}}\). A decay, further referred as fictitious fatigue, can be observed for all the values of the regularization parameter plotted. This behavior becomes more clear as \(\lambda \) increases. In order to quantify the fictitious fatigue, the value at each peak is compared to the value at \(t_{\textrm{t}}\) through the measure

$$\begin{aligned} m_{\textrm{f}} {:}{=} 1 - \dfrac{{\hat{T}}_{12}\left( t_{\textrm{t}} + n\tau ; {{\,\textrm{sig}\,}}_{\lambda }\right) }{\max \limits _{t \in \left[ 0,t_{\textrm{t}} + n\tau \right] } \left( {\hat{T}}_{12} \left( t; {{\,\textrm{sig}\,}}_{\lambda }\right) \right) }, \end{aligned}$$
(26)

which is plotted in Fig. 7 for all sigmoid functions with \(\lambda = {}{e-8}{}\). Following the trend of the previous subsection, \({{\,\textrm{sig}\,}}_{\lambda }^{\textrm{er}}\), \({{\,\textrm{sig}\,}}_{\lambda }^{\textrm{th}}\) and \({{\,\textrm{sig}\,}}_{\lambda }^{\textrm{gd}}\) provide the best results with similar values of \(m_{\textrm{f}}\). Out of the three, \({{\,\textrm{sig}\,}}_{\lambda }^{\textrm{er}}\) introduces the least fictitious fatigue. A considerable increase in values is showcased by \({{\,\textrm{sig}\,}}_{\lambda }^{\textrm{sq}}\) and \({{\,\textrm{sig}\,}}_{\lambda }^{\textrm{at}}\), specially the latter. All curves run fairly parallel to each other with a clear limit to infinity, i.e., the fictitious fatigue becomes greater with each increase of the number of cycles. The values of \(m_{\textrm{f}}\) for all \({{\,\textrm{sig}\,}}_{\lambda }^{\textrm{er}}\) are plotted in Fig. 8. Apart from the \(\lambda ={}{e-3}{}\) curve, the curves not only run considerably parallel to each other but they showcase a fairly linear behavior with respect to \(\lambda \) for a given peak number of cycles, which becomes clearer by exchanging the role of n and \(\lambda \) while plotting, as done in Fig. 9. The rate-dependency introduced by the regularization is further confirmed by plotting \(m_{\textrm{f}}\) for different period length values, as seen in Fig. 10 for \({{\,\textrm{sig}\,}}_{{}{e-7}{}}^{\textrm{er}}\) where longer processes lead to a more significant fictitious fatigue.

Fig. 7
figure 7

Evaluation of \(m_{\textrm{f}}\) for different sigmoid functions with \(\lambda ={}{e-8}{}\)

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Fig. 8
figure 8

Evaluation of \(m_{\textrm{f}}\) using \({{\,\textrm{sig}\,}}_{\lambda }^{\textrm{er}}\) for different values of the regularization parameter

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Fig. 9
figure 9

Evaluation of \(m_{\textrm{f}}\) using \({{\,\textrm{sig}\,}}_{\lambda }^{\textrm{er}}\) for different values of the regularization parameter

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Fig. 10
figure 10

Evaluation of \(m_{\textrm{f}}\) using \({{\,\textrm{sig}\,}}_{\lambda }^{\textrm{er}}\) with \(\lambda = {}{e-7}{}\) for different period length

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5 Conclusions

The numerical benefits of recurring to the viscoplastic regularization come at the cost of introducing undesired rate-dependent behavior. In the context of the employed gradient crystal plasticity formulation, this is manifested in the continuous evolution of the plastic slips. In turn, the constant plastic flow can lead to effects that can be mistakenly perceived as physical phenomena, e.g., material softening and fatigue. Although it is evident that these fictitious effects are reduced as the rate-independent case is more accurately replicated by the viscoplastic regularization, there is a numerical limit due to the intrinsic non-differentiable transition of an elastic process becoming inelastic. This numerical limit is further influenced by the complexity of the mathematical model and loads. As such, the knowledge of which fictitious behavior may be at play and to what extent is important while analyzing the numerical results. By solving a simple boundary value problem the present work aims to provide a sense of the magnitude in which these formulation artifacts can affect the solution.

When considering a static load, the constant plastic flow is perceived as an fictitious softening of the material, as the stress decays while the strain load increases. In the boundary value problem solved, the results show that the load at which the softening starts is inversely proportional to the regularization parameter. Although the softening is negligible for most of the sigmoid functions and values of the regularization parameter tested, it reaches quite noticeable effects for some of the functions using the highest parameter value. Therefore, there is a maximum value of the regularization parameter allowed for a given softening tolerance.

The constant plastic flow becomes particularly problematic when considering cyclic loads. Theoretically, the fictitious fatigue will become noticeable for any value of the regularization parameter if the amount of cycles is high enough. As such, the regularization parameter should be determined as inversely proportional to the amount of cycles planned to simulate. In the case of the boundary value problems solved in the present work, a relation between the decay and the number of cycles and the regularization parameter can be extracted from the obtained data. Hinting that this may be possible for other simple boundary value problems.

In summary, the admissible values of the regularization parameter depends on the type of loading, being more restrictive for cyclic loads, and the amount of fictitious effects that can be tolerated in the results.

The choice of the sigmoid function was shown to be relevant as some achieve acceptable results with higher values of the regularization parameter while other require considerable smaller values to obtain similar results. Among the sigmoid functions chosen, the ones based on the error function, the hyperbolic tangent and the Gudermannian function provide the best approximations, in that specific order. Whenever possible the sigmoid function based on the error function should be implemented. The authors advise against the use of the sigmoid functions formulated around the arctangent and square root functions, as they perform poorly when compared to the rest.