1 Introduction

The need for uncertainty quantification methods in engineering simulations led to the development of many different methods in the framework of stochastic finite element methods. The methods were developed first for linear elastic material behavior. An overview of stochastic finite element methods is presented in, e.g., [2, 28]. Unfortunately, the application of these methods to nonlinear and/or inelastic material is not trivial. Significant progress was mainly made for the special cases of hyperelasticity [18, 24] and plasticity [1, 22, 25, 32]. For more general inelastic behavior, three main methods can be differentiated:

Sampling:

The classical Monte Carlo scheme ([13, 21]) is most likely the best known sampling method. Repeated evaluation with various realizations of the random parameters allows to approximate the response characteristics. While being the most robust method, it comes with a significant computational effort as many individual simulations have to be performed. In fact, application to industrial problems is prohibited due to extensive numerical costs.

Collocation:

A clever choice of the sampling points, e.g., Smolyak points ([31]), reduces the number of individual simulations needed to be performed. A reduced-order polynomial surface is used for interpolation in between sampling points [9, 15, 30]. Nevertheless, especially, in the high-dimensional case the number of collocation points can blow up quickly. Furthermore, instability is a reoccurring problem [7].

Perturbation:

Perturbation methods [20] are the most computationally efficient strategies. Here, a Taylor series is used to approximate the quantities of interest around the expectation of the random variables. Unfortunately, this method can only be used for simplified (polynomial) material models reducing the overall predictability capability of the method. Applications can be found in, e.g., [8, 29].

It is obvious that there exists a need for a computationally efficient yet accurate method. The time-separated stochastic mechanics (TSM) [17] is a novel method to fill this research gap. First developed for structures with viscoelastic materials, it showed surprisingly good results for its low computational footprint. The method is based on a Taylor series representation of all fields as displacements and internal variables in the random variables. Such, it approximates the time-dependent and stochastic fields by a sum of products of time-dependent but deterministic and stochatic but time-independent terms. The time-dependent terms are calculated by a low number of standard deterministic finite element simulations. The effect of the fluctuations of the random variables is introduced in a post-processing step. In contrast to the perturbation method, it can handle inelastic material models with internal variables directly. Indeed, a main advantage of the TSM is that evolution laws for the internal variable in form of differential equations can be handled. For homogeneous material fluctuations, only one single extended deterministic FEM simulation was needed to accurately predict expectation and standard deviation of stress and reaction force. Since then, the TSM was used in a variety of other problems, e.g., dynamic simulations [11] and local material fluctuations [10]. However, up to now only a viscoelastic material model was investigated. Recently, it was shown that the approach can be extended to a larger class of material models [12] on the material point level. In this work, we are building up on the work of [12] and present the TSM for uncertainty quantification of structures with elasto-viscoplastic material behavior.

1.1 Contribution

In this work, we present a method for efficient and accurate uncertainty quantification for inelastic engineering simulations. It allows to quickly assess expectation and standard deviation of stress, internal variable and reaction force as only a low number of deterministic FEM simulations and several fast matrix multiplications are needed. The method is based on a separation of the fields of strain and internal variable(s) into stochastic but time-independent and deterministic but time-dependent parts. This allows to effectively split the time-dependent from the stochastic behavior for all equations, i.e., balance of linear momentum and evolution equation. A main novelty of the method is the approximation of the evolution equation which is typically only available as a differential equation. This, in return, allows to combine a high computational efficiency with great accuracy. The method is presented for the case of an elasto-viscoplastic material behavior which posed severe problems for uncertainty quantification up to now. Comparisons with the standard Monte Carlo method confirm the accuracy and efficiency for a variety of boundary value problems.

2 Karhunen–Loéve expansion

As already established, all material parameters are subject to random fluctuations. An important modeling assumption is whether the fluctuations are homogeneous over the whole structure or if the parameters fluctuate locally. One reason for homogeneous fluctuations can be accidental variations in the production process. Microstructural material irregularities may lead to local fluctuations. Of course, the boundary between the two types is rather blurry. As the parameters are random functions which take a random value at each point \(\varvec{x}\) in the domain, they are modeled as a Gaussian random field \(H(\varvec{x}, \theta )\). It is important to mention that the modeling as a Gaussian random field allows—in principle—for unphysical negative material parameters. To avoid this case, we employ the strategy presented in Sect. 5. The variable \(\theta \) encodes all characterizing (and typically unknown) information about the individual material specimen. A Gaussian random field is defined by its mean and its covariance function. The mean function defines the expectation of the field for each point in the domain and is given as

$$\begin{aligned} \langle H(\varvec{x}) \rangle = \int _{\Omega } H(\varvec{x}, \theta ) \,\mathbb {P}(\textrm{d}\theta ) =: \bar{H}(\varvec{x}) \end{aligned}$$
(1)

where \(\mathbb {P}\) is the Gaussian probability density function. The covariance function encodes the spatial correlation, e.g., to what degree the fluctuations in nearby points are correlated. A covariance function between two points \(\varvec{x}\) and \(\varvec{x}'\) is given as

$$\begin{aligned} \text {Cov}(\varvec{x}, \varvec{x}') = \text {Std}(\varvec{x})\text {Std}(\varvec{x}')\rho (\varvec{x}, \varvec{x}'). \end{aligned}$$
(2)

Here, \(\text {Std}(\varvec{x})\) denotes the standard deviation of the Gaussian random field at a point and \(\rho (\varvec{x}, \varvec{x}')\) a correlation function. The specification of the correlation function influences the correlation structure and is a user choice. Consequently, various different correlation functions have been proposed, cf. [14, 27]. Here, the exponential correlation function in the form

$$\begin{aligned} \rho (\varvec{x}, \varvec{x}') = \exp (-||\varvec{x}-\varvec{x}'||/l) \end{aligned}$$
(3)

is used. For a correlation length \(l \rightarrow \infty \) a homogeneous material fluctuation results, i.e., all points in the domain fluctuate simultaneously. The discretization of the random field remains to be discussed. Equivalently to the space and time discretization, this step can be seen as a discretization of the stochastic dimension. A widely used discretization is the Karhunen–Loéve expansion [19, 23]. It represents the random field as a linear combination of a set of orthogonal basis functions as

$$\begin{aligned} H(\varvec{x}, \theta ) = \bar{H}(\varvec{x}) + \sum _{i=1}^\infty \xi ^i(\theta ) \sqrt{k^i}\varphi ^i(\varvec{x}) = \bar{H}(\varvec{x}) + \sum _{i=1}^\infty \xi ^i(\theta )H^i(\varvec{x}). \end{aligned}$$
(4)

Here, \(\xi (\theta )\) are standard uncorrelated random variables and \(k^i\) and \(\varphi ^i\) are obtained by the solution of the homogeneous Fredholm integral equation of the second kind

$$\begin{aligned} \int _{\Omega } \text {Cov}(\varvec{x}, \varvec{x}') \, \varphi ^i(\varvec{x}') \,\textrm{d}\varvec{x}' = k^i \varphi ^i(\varvec{x}). \end{aligned}$$
(5)

Solution techniques for Eq.  (5) are discussed among others in [4, 16] or [3]. Since the Karhunen–Loéve expansion in Eq.  (4) has infinite terms, it has to be truncated for applications as finite element simulations. The resulting truncation error if only the first M terms are used is given for a constant standard deviation of the field \(\varsigma \) as

$$\begin{aligned} \text {err}^\rho = 1-\frac{1}{\Omega } \frac{1}{\varsigma ^2}\sum _{i=1}^M k^i \end{aligned}$$
(6)

with the volume \(\Omega \).

3 Overview of the method

Inelastic material models are typically modeled using internal variables which incorporate information on the micromechanical state. The evolution of the internal variable is described by an evolution equation often in the form of a differential equation. The main problem of approximating inelastic material behavior under uncertainty stems from the fact that the uncertain parameters influence both the evolution of the internal variable and the strain field. In addition, the interaction of all fields rises the overall complexity even further. Therefore, a suitable approximation must be skillfully designed. The time-separated stochastic mechanics was developed exactly for this purpose: approximating the behavior of structures with inelastic materials with the lowest possible number of finite element simulations. The main idea is the expansion of all fields, i.e., strain field and internal variable(s) in a stochastic series expansion. This allows to easily track the effect of the fluctuations of material parameters on these fields. Careful examination of the governing equations, i.e., balance of linear momentum and evolution equation, allows to separate stochastic from time-dependent behavior. Such, only the time-dependent behavior is simulated. The effect of the fluctuations of the material parameters is included during post-processing. Each term of the series expansion of the strain field corresponds to one additional balance equation which has to be solved via, e.g., finite element simulations. In contrast, each term of the series expansion of the internal variable leads to one additional internal variable with its own evolution equation. It may be remarked that all equations to calculate the terms in the series expansions can be easily determined by examination of the standard deterministic balance of linear momentum and evolution equation.

In the following, the approach is explained in detail. This includes the material model and modeling assumptions in Sect. 3.1, the series expansions in Sect. 3.2 and the determination of the system to be solved, i.e., balance equations, cf. Section 3.3 and evolution laws, cf., Sect. 3.4.

3.1 Material model and modeling assumptions

As a prototype material model used in engineering simulations, we are investigating an elasto-viscoplastic material. It shows rate-independent as rate-dependent characteristics. Furthermore, viscoelasticity and plasticity are included as special cases. These effects and the underlying mathematical structure make the investigation of structures with elasto-viscoplasticity very interesting. With the usual introduction of an additional internal variable \(\varvec{\varepsilon }^{\text {vp}}\) capturing the viscoplastic strains, the stress \(\varvec{\sigma }\) may be written as

$$\begin{aligned} \varvec{\sigma }= \mathbb {E}\cdot (\varvec{\varepsilon }- \varvec{\varepsilon }^{\text {vp}})\,. \end{aligned}$$
(7)

Here, \(\mathbb {E}\) denotes the elasticity tensor and \(\varvec{\varepsilon }\) the total strains. The evolution of the internal variable is described by a differential algebraic equation

$$\begin{aligned} \dot{\varvec{\varepsilon }}^{\text {vp}} = \eta ^{-1} \left( ||\textrm{dev}(\varvec{\sigma })|| - \sigma ^\text {Y} \right) _+ \frac{\textrm{dev}(\varvec{\sigma })}{||\textrm{dev}(\varvec{\sigma })||} \end{aligned}$$
(8)

with the viscosity \(\eta \), the scalar yield limit \(\sigma ^Y\) and the Macauley brackets defined as \((x)_+ = \text {max}(0, x)\). The deviatoric part of the stress is calculated as usual as \(\textrm{dev}(\varvec{\sigma }) = \varvec{\sigma }- \frac{1}{3}(\sigma _1 + \sigma _2 + \sigma _3) \mathbb {I}= \mathbb {S}\cdot \varvec{\sigma }\) with the identity tensor \(\mathbb {I}\). For a shorter notation, the linear operator \(\mathbb {S}\) is introduced in Voigt notation as

$$\begin{aligned} \mathbb {S}= \begin{pmatrix} \frac{2}{3} &{}\quad -\frac{1}{3} &{}\quad -\frac{1}{3} &{}\quad 0 &{}\quad 0 &{}\quad 0\\ -\frac{1}{3} &{}\quad \frac{2}{3} &{}\quad -\frac{1}{3} &{}\quad 0 &{}\quad 0 &{}\quad 0\\ -\frac{1}{3} &{}\quad -\frac{1}{3} &{}\quad \frac{2}{3} &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0&{}\quad 0 &{}\quad 0&{}\quad 1 &{}\quad 0 &{}\quad 0 \\ 0&{}\quad 0 &{}\quad 0&{}\quad 0 &{}\quad 1 &{}\quad 0 \\ 0&{}\quad 0 &{}\quad 0&{}\quad 0 &{}\quad 0 &{}\quad 1 \\ \end{pmatrix} \end{aligned}$$
(9)

The material parameters of the model are the viscosity \(\eta \), the scalar yield limit \(\sigma ^\text {Y}\) and for an isotropic material the Lamé parameters \(\lambda \) and \(\mu \) which make up the elasticity tensor. It remains to the user to find suitable modeling assumptions concerning the uncertainty of these parameters. While experiments are the gold standard, a large number is typically needed to accurately identify the probability distribution of the parameters. As these are material parameters they are independent of geometry and loading scenario. Often heuristic assumptions are made. Here, we assume that \(\lambda \) and \(\mu \) are fully stochastically dependent and that the viscosity parameter is deterministic. It may be remarked that the method is well able to incorporate any assumption on the randomness of the material parameters. With these assumptions, the elasticity tensor can be expanded in the expansion

$$\begin{aligned} \mathbb {E}(\varvec{x}, \theta ) = \mathbb {E}^0(\varvec{x}) + \xi ^{\mathbb {E}, i}(\theta ) \mathbb {E}^i(\varvec{x}) \end{aligned}$$
(10)

and the yield limit as

$$\begin{aligned} \sigma ^{\text {Y}}(\varvec{x}, \theta ) = \sigma ^{\text {Y}, 0} + \xi ^{\text {Y}, i} \sigma ^{\text {Y}, i} \end{aligned}$$
(11)

for every point \(\varvec{x}\in \Omega \). Here and in the following, we are avoiding the summation sign over the Karhunen–Loéve terms i for a clearer presentation.

3.2 Series expansions

Subsequently, each field is expanded as a linear series expansion in the random variables of the Karhunen–Loéve expansions. We denote this approximation with the upper index TSM. Such, the displacement field reads as

$$\begin{aligned} \varvec{u}^\text {TSM}({t, } \varvec{x}, \theta ) = \varvec{u}^0({t, }\varvec{x}) + \xi ^{\mathbb {E}, i}(\theta ) \varvec{u}^{\mathbb {E}, i}({t, }\varvec{x}) + \xi ^{\text {Y}, j}(\theta ) \varvec{u}^{\text {Y}, j}({t, }\varvec{x}) \approx \varvec{u}({t, }\varvec{x}, \theta ) \end{aligned}$$
(12)

and consequently the strain field is given by \(\varvec{\varepsilon }= \nabla \varvec{u}\) as

$$\begin{aligned} \varvec{\varepsilon }^\text {TSM}({t, }\varvec{x}, \theta ) = \varvec{\varepsilon }^0({t, }\varvec{x}) + \xi ^{\mathbb {E}, i}(\theta ) \varvec{\varepsilon }^{\mathbb {E}, i}({t, }\varvec{x}) + \xi ^{\text {Y}, j}(\theta ) \varvec{\varepsilon }^{\text {Y}, j}({t, }\varvec{x}) \approx \varvec{\varepsilon }({t, }\varvec{x}, \theta ). \end{aligned}$$
(13)

The series expansion as above effectively separates the time-dependent but deterministic terms (\(\xi ^{\mathbb {E}, i}\) and \(\xi ^{\textrm{Y}, i}\)) from stochastic but time-independent terms. This separation gives our method its name: Time-Separated Stochastic Mechanics (TSM). Of course, higher-order terms can be used to increase the accuracy at the cost of computational effort or for error estimation. Thus, comparison of results with varying numbers of TSM terms enables a quantification of the error. Then, route correction methods could be implemented, e.g., an adaptive number of TSM terms. This property of TSM will be investigated in future work. However, for the investigated problems, a linear Taylor series suffices. On the contrary, the computational cost may be reduced by neglecting the influence of a random variable on the strain. The same expansion is also used for the internal variable as

$$\begin{aligned} \varvec{\varepsilon }^{\text {vp}, \text {TSM}}({t, }\varvec{x}, \theta ) = \varvec{\varepsilon }^{\text {vp}, 0}({t, }\varvec{x}) + \xi ^{\mathbb {E}, i}(\theta ) \varvec{\varepsilon }^{\text {vp}, \mathbb {E}, i}({t, }\varvec{x}) + \xi ^{\text {Y}, j}(\theta ) \varvec{\varepsilon }^{\text {vp}, \text {Y}, j}({t, }\varvec{x}) \approx \varvec{\varepsilon }^{\text {vp}}({t, }\varvec{x}, \theta ). \end{aligned}$$
(14)

With these expansions, the expression for the stress can be similarly ordered as

$$\begin{aligned} \varvec{\sigma }{(t, \varvec{x}, \xi )}&= \mathbb {E}^0\cdot (\varvec{\varepsilon }^0{(t, \varvec{x})} - \varvec{\varepsilon }^{\text {vp}, 0}{(t, \varvec{x})}) \nonumber \\&+ \left( \mathbb {E}^i \cdot (\varvec{\varepsilon }^0{(t, \varvec{x})} - \varvec{\varepsilon }^{\text {vp}, 0}{(t, \varvec{x})}) + \mathbb {E}^0 \cdot (\varvec{\varepsilon }^{\mathbb {E}, j}{(t, \varvec{x})} - \varvec{\varepsilon }^{\text {vp}, \mathbb {E}, j}{(t, \varvec{x})}) \right) \xi ^{\mathbb {E}, i} \nonumber \\&+ \mathbb {E}^0 \cdot (\varvec{\varepsilon }^{\text {Y}, k}{(t, \varvec{x})} - \varvec{\varepsilon }^{\text {vp}, \text {Y}, k}{(t, \varvec{x})})\xi ^{\text {Y},k}. \end{aligned}$$
(15)

3.3 Balance equations

With the previous expansion of the stress, the usual balance of linear momentum in variational form reads as

$$\begin{aligned} \int _\Omega \delta \varvec{\varepsilon }^0 \cdot (\mathbb {E}^0&+ \xi ^{\mathbb {E}, i}\mathbb {E}^i) \cdot (\varvec{\varepsilon }^{\text {el}, 0} + \xi ^{\mathbb {E}, j} \varvec{\varepsilon }^{el, \mathbb {E}, j} + \xi ^{\text {Y}, k} \varvec{\varepsilon }^{\text {el},\text {Y},k}) \,\textrm{d}V \nonumber \\&- \int _{\Omega } \delta \varvec{u}^0\cdot \varvec{b}^* \,\textrm{d}V - \int _{\partial \Omega } \delta \varvec{u}^0\cdot \varvec{t}^* \,\textrm{d}A = 0 \quad \forall \delta \varvec{\varepsilon }^0 \quad \forall \delta \varvec{u}^0. \end{aligned}$$
(16)

Here, the elastic part of the strains defined as \(\varvec{\varepsilon }^{\text {el}} = \varvec{\varepsilon }- \varvec{\varepsilon }^{\text {vp}}\) is introduced for a shorter notation. By ordering Eq.  (16) by the random terms, three independent equations can be identified as

$$\begin{aligned}&\int _{\Omega } \delta \varvec{\varepsilon }^0\cdot \mathbb {E}^0\cdot \varvec{\varepsilon }^{\text {el}, 0} \,\textrm{d}V - \int _{\Omega }\delta \varvec{u}^0\cdot \varvec{b}^* \,\textrm{d}V - \int _{\partial \Omega } \varvec{u}^0 \cdot \varvec{t}^* \,\textrm{d}A = 0 \end{aligned}$$
(17)
$$\begin{aligned}&\int _\Omega \delta \varvec{\varepsilon }^0\cdot (\mathbb {E}^i \cdot \varvec{\varepsilon }^{\text {el}, 0} + \mathbb {E}^0 \cdot \varvec{\varepsilon }^{\text {el}, \mathbb {E}, i}) \,\textrm{d}V = 0 \end{aligned}$$
(18)
$$\begin{aligned}&\int _\Omega \delta \varvec{\varepsilon }^0 \cdot (\mathbb {E}^0 \cdot \varvec{\varepsilon }^{\text {el}, \text {Y}, i}) \,\textrm{d}V = 0 \end{aligned}$$
(19)

These equations constitute the three individual balances that have to be fulfilled in order to identify the factors in the corresponding Taylor series. By Eq.  (17), the standard deterministic strain \(\varvec{\varepsilon }^0\) can be identified. With Eqs.  (18) and (19), the terms \(\varvec{\varepsilon }^{\mathbb {E}, i}\) and \(\varvec{\varepsilon }^{\text {Y}, i}\) are calculated. It may be remarked that Eq.  (18) requires the solution of Eq.  (17). Thus, a staggered solution scheme is the automatic consequence of our approach with TSM series representations. These represent the (linear) change of the strain field due to changes in the elasticity tensor and the yield limit, respectively. Each of those terms is dependent on the term i of the Karhunen–Loéve series. This is intuitively comprehensible since with each spatial fluctuation of the material parameters, the strain field changes.

3.4 Evolution of the internal variable

The evolution of the internal variable is directly linked to the evolution of the strain field. In fact, the evolution of the internal variable and the strain field are influencing each other mutually. Therefore, the changes of the evolution of the internal variable have to be approximated. The Taylor series expansion in Eq.  (14) enables exactly this. However, the terms \(\varvec{\varepsilon }^{\text {vp}, \mathbb {E}, i}(t)\) and \(\varvec{\varepsilon }^{\text {vp}, \text {Y}, j}(t)\) tracking the effect of fluctuations in \(\mathbb {E}\) and \(\sigma ^\text {Y}\) have to be determined. This presents a major challenge since (1) these terms are time-dependent, and (2) the evolution of the viscoplastic strain is only known in the form of a differential equation, i.e., \(\dot{\varvec{\varepsilon }}^\text {vp} = f(\varvec{\varepsilon }^\text {vp}, \mathbb {E}, \sigma ^\text {Y})\). Consequently, the evolution of \(\varvec{\varepsilon }^{\text {vp}, \mathbb {E}, i}(t)\) and \(\varvec{\varepsilon }^{\text {vp}, \text {Y}, j}(t)\) have to be extracted out of the differential equation. For this, we make use of the approach as presented in [12].

The basic idea of the approach is to use the approximation of the internal variable in Eq.  (14) in evolution Eq. (8) as

$$\begin{aligned} \dot{\varvec{\varepsilon }}^\text {vp, TSM} = f(\varvec{\varepsilon }^\text {vp, TSM}, \mathbb {E}, \sigma ^\text {Y}). \end{aligned}$$
(20)

Here, the Karhunen–Loéve expansion for the elasticity tensor and the yield limit in Eqs.  (10) and (11) are used.

The standard deterministic evolution equation can be recovered from the evaluation of Eq.  (20) at \(\xi ^{\mathbb {E}, i} = \xi ^{\text {Y}, j} = 0\) as

$$\begin{aligned} \dot{\varvec{\varepsilon }}^{\text {vp}, 0}(t) = \dot{\varvec{\varepsilon }}^\text {vp, TSM}(t)\big |_{\xi ^{\mathbb {E}, i} = \xi ^{\text {Y}, j} = 0}. \end{aligned}$$
(21)

We find the evolution equation for the term \(\dot{\varvec{\varepsilon }}^{\text {vp}, \mathbb {E}, i}\) by taking the derivative of the evolution equation (cf. Eq.  (8)) for each component \(\alpha \) in the Gâteaux sense as

$$\begin{aligned} \dot{\varvec{\varepsilon }}^{\text {vp}, \mathbb {E}, i}_\alpha (t)&= \frac{\textrm{d}\dot{\varvec{\varepsilon }}^{\text {vp, TSM}}_\alpha (t)}{\textrm{d}\xi ^{\mathbb {E}, i}}\Big |_{\xi ^{\mathbb {E}, j} = \xi ^\textrm{Y, k}=0} \quad \nonumber \\&= \frac{1}{\eta } \bigg ( H\left( ||\text {dev}(\varvec{\sigma })||-\sigma ^\text {Y}\right) \frac{\text {dev}(\varvec{\sigma })_d}{||\text {dev}(\varvec{\sigma })||} \frac{\textrm{d}}{\textrm{d}\xi ^{\mathbb {E}, i}} (\text {dev}(\varvec{\sigma })_d)\frac{\text {dev}(\varvec{\sigma })_\alpha }{||\text {dev}(\varvec{\sigma })||} \nonumber \\&\quad +\left( ||\text {dev}(\varvec{\sigma })|| - \sigma ^\text {Y}\right) _+ \frac{1}{||\text {dev}(\varvec{\sigma })||} \frac{\textrm{d}}{\textrm{d}\xi ^{\mathbb {E}, i}} (\text {dev}(\varvec{\sigma })_\alpha ) + \text {dev}(\varvec{\sigma })_\alpha \frac{\textrm{d}}{\textrm{d}\xi ^{\mathbb {E}, i}} \left( || \text {dev}(\varvec{\sigma }) ||^{-1} \right) \bigg )\Big |_{\xi ^{\mathbb {E}, j} = \xi ^\textrm{Y, k}=0}, \end{aligned}$$
(22)

evaluated at \(\xi ^{\mathbb {E}, j} = \xi ^\textrm{Y, k}=0\). Here, H(x) denotes the Heaviside step function. Since \(\alpha \) is a fixed component, no summation is carried out over this index. The remaining derivatives are found as

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}\xi ^{\mathbb {E}, i}} (\text {dev}(\varvec{\sigma })_d) = \mathbb {S}\cdot \mathbb {E}^i \cdot (\varvec{\varepsilon }^0 - \varvec{\varepsilon }^{\textrm{vp}, 0}) - \mathbb {S}\cdot \mathbb {E}^0 \cdot \varvec{\varepsilon }^{\textrm{vp}, \mathbb {E}, i} \end{aligned}$$
(23)

and

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}\xi ^{\mathbb {E}, i}} (||\text {dev}(\varvec{\sigma })||^{-1}) = - ||\text {dev}(\varvec{\sigma })||^{-2} \frac{\text {dev}(\varvec{\sigma })_d}{||\text {dev}(\varvec{\sigma })||} \frac{\textrm{d}}{\textrm{d}\xi ^{\mathbb {E}, i}} (\text {dev}(\varvec{\sigma })_d). \end{aligned}$$
(24)

The evolution equation for the term \(\dot{\varvec{\varepsilon }}^{\text {vp}, \mathbb {E}, i}\) requires the solution for \(\varvec{\varepsilon }^0\) and \(\varvec{\varepsilon }^{\text {vp}, 0}\) first. Thus, a staggered scheme results.

Similarly, the evolution equation for the term \(\dot{\varvec{\varepsilon }}^{\text {vp}, \text {Y}, i}\) is found using the Gâteaux derivative with respect to the random variable for the yield limit as

$$\begin{aligned} \dot{\varvec{\varepsilon }}^{\text {vp}, \text {Y}, j}_\alpha (t)&= \frac{\textrm{d}\dot{\varvec{\varepsilon }}^{\textrm{vp, TSM}}_\alpha (t)}{\textrm{d}\xi ^{\textrm{Y}, j}} \Big |_{\xi ^{\mathbb {E}, j} = \xi ^\textrm{Y, k}=0} \nonumber \\&= \frac{1}{\eta } \bigg ( H(||\text {dev}(\varvec{\sigma })|| - \sigma ^{\textrm{Y}, 0}) \left( \frac{\text {dev}(\varvec{\sigma })_d}{||\text {dev}(\varvec{\sigma })||} \frac{\textrm{d}}{\textrm{d}\xi ^{\textrm{Y}, j}} (\text {dev}(\sigma )_d) - 1 \right) \frac{\text {dev}(\varvec{\sigma })_\alpha }{||\text {dev}(\varvec{\sigma })||} \nonumber \\&\quad + \left( ||\text {dev}(\varvec{\sigma })|| - \sigma ^{\textrm{Y}, 0} \right) _+ \frac{\textrm{d}}{\textrm{d}\xi ^{\textrm{Y}, j}} \Big ( \frac{\text {dev}(\varvec{\sigma })_\alpha }{||\text {dev}(\varvec{\sigma })||} \Big ) \bigg )\Big |_{\xi ^{\mathbb {E}, j} = \xi ^\textrm{Y, k}=0} \end{aligned}$$
(25)

with the quantities

$$\begin{aligned}{} & {} \frac{\textrm{d}}{\textrm{d}\xi ^{\textrm{Y}, j}} \Big (\text {dev}(\varvec{\sigma })_d \Big ) = - \mathbb {S}\cdot \mathbb {E}^0 \cdot \varvec{\varepsilon }^{\textrm{vp}, \textrm{Y}, j}, \end{aligned}$$
(26)
$$\begin{aligned}{} & {} \frac{\textrm{d}}{\textrm{d}\xi ^{\textrm{Y}, j}} \Big ( \frac{\text {dev}(\varvec{\sigma })_\alpha }{||\text {dev}(\varvec{\sigma })||} \Big ) = \frac{1}{||\text {dev}(\varvec{\sigma })||} \frac{\textrm{d}}{\textrm{d}\xi ^{\textrm{Y}, j}}(\text {dev}(\varvec{\sigma })_\alpha ) + \text {dev}(\varvec{\sigma })_\alpha \frac{\textrm{d}}{\textrm{d}\xi ^{\textrm{Y}, j}} (|| \text {dev}(\varvec{\sigma })||^{-1} ) \end{aligned}$$
(27)

and

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}\xi ^{\textrm{Y}, j}} (|| \text {dev}(\varvec{\sigma })||^{-1} ) = - ||\text {dev}(\varvec{\sigma })||^{-2} \frac{\text {dev}(\varvec{\sigma })_d}{||\text {dev}(\varvec{\sigma })||} \frac{\textrm{d}}{\textrm{d}\xi ^{\textrm{Y}, j}} \Big (\text {dev}(\varvec{\sigma })_d \Big ). \end{aligned}$$
(28)

4 Uncertainty quantification

For an engineering simulation, the uncertainty in stress and reaction force are important to be quantified in order to assess reliability and design weaknesses. Additionally, it may be beneficial to approximate the uncertainty in the internal variable, in this example it is the viscoplastic strains. All of these quantities are time-dependent and dependent on the loading. Naturally, also the uncertainty in these quantities is time-dependent. A simple way to quantify the uncertainty in an output quantity is to present expectation and standard deviation. The expectation or expected value represents the average value for a large number of random realizations of the input random variables. The standard deviation is a measure of the spread of the output quantity. A large standard deviation therefore hints at a large uncertainty. Furthermore, these quantities directly allow for reliability assessments by the Chebyshev inequality. It states that the probability for a quantity x being more than k-times the standard deviation distant from the expectation is upper bounded by

$$\begin{aligned} \int _{\langle x \rangle + k\, \text {Std}(x)}^{\infty } x \ \mathbb {P}(\textrm{d}\theta ) \le \frac{1}{2 \, k^2}. \end{aligned}$$
(29)

For example, the probability that \(x \ge \langle x \rangle + 2 \, \text {Std}(x)\) is less than \(12.5\%\).

In the following, the calculation of the expectation and standard deviation of internal variable, stress and reaction force are presented.

4.1 Expectation and variance of the internal variable \(\varvec{\varepsilon }^{\text {vp}}\)

The calculation of the expectation of the internal variable is straightforward. With the series expansion of Eq.  (14), the expectation is given by

$$\begin{aligned} \langle \varvec{\varepsilon }^\text {vp}(t) \rangle = \varvec{\varepsilon }^{\text {vp}, 0}(t) \end{aligned}$$
(30)

The variance of each component \(\alpha \) of the viscoplastic strains is calculated by using the relation

$$\begin{aligned} \text {Var}(\varepsilon _\alpha ^{\text {vp}}) = \langle \left( \varvec{\varepsilon }^{\text {vp}} \right) ^2 \rangle -\langle \left( \varvec{\varepsilon }^{\text {vp}} \right) \rangle ^2. \end{aligned}$$
(31)

It remains to calculate the second moment \(\langle \left( \varvec{\varepsilon }^{\text {vp}}\right) ^2 \rangle \). The second moment for each component \(\alpha \) is given as

$$\begin{aligned} \langle (\varepsilon ^{\text {vp}}_\alpha (t))^2 \rangle = (\varepsilon _\alpha ^{\text {vp}, 0}(t))^2 + (\varepsilon _\alpha ^{\text {vp}, \mathbb {E}, i}(t))^2 + (\varepsilon _\alpha ^{\text {vp}, \text {Y}, j}(t))^2 + \varepsilon _\alpha ^{\text {vp}, \mathbb {E}, k}(t) \varepsilon _\alpha ^{\text {vp}, \text {Y}, l}(t) \langle \xi ^{\mathbb {E}, k} \xi ^{\text {Y}, l} \rangle . \end{aligned}$$
(32)

We remark that due to the mathematical structure of the Karhunen–Loéve expansion \(\langle (\xi ^{\mathbb {E}, i})^2 \rangle = \langle (\xi ^{\text {Y}, i})^2 \rangle = 1\), i.e., they are standard normal random variables. The term \(\langle \xi ^{\mathbb {E}, k}, \xi ^{\text {Y}, k} \rangle \) denotes the stochastic coupling of the fluctuations and is given by the modeling assumptions.

4.2 Expectation and variance of the stress

The stress is given by Eq.  (15) and recalled here for convenience as

$$\begin{aligned} \varvec{\sigma }= \mathbb {E}^0\cdot (\varvec{\varepsilon }^0 - \varvec{\varepsilon }^{\text {vp}, 0}) + \left( \mathbb {E}^i \cdot (\varvec{\varepsilon }^0 - \varvec{\varepsilon }^{\text {vp}, 0}) + \mathbb {E}^0 \cdot (\varvec{\varepsilon }^{\mathbb {E}, j} - \varvec{\varepsilon }^{\text {vp}, \mathbb {E}, j}) \right) \xi ^{\mathbb {E}, i} + \mathbb {E}^0 \cdot (\varvec{\varepsilon }^{\text {Y}, k} - \varvec{\varepsilon }^{\text {vp}, \text {Y}, k})\xi ^{\text {Y},k}. \end{aligned}$$
(33)

The expectation of the stress is trivially found as

$$\begin{aligned} \langle \varvec{\sigma }\rangle = \mathbb {E}^0\cdot (\varvec{\varepsilon }^0 - \varvec{\varepsilon }^{\text {vp}, 0}). \end{aligned}$$
(34)

For the calculation of the variance, we again make use of the relation

$$\begin{aligned} \text {Var}(\sigma _\alpha ) = \langle \sigma _\alpha ^2 \rangle - \langle \sigma _\alpha \rangle ^2 \end{aligned}$$
(35)

for each component \(\alpha \). The second moment \(\langle \sigma _\alpha ^2 \rangle \) for each component \(\alpha \) is given as

$$\begin{aligned} \langle \sigma _\alpha ^2 \rangle&= \langle \sigma _\alpha \rangle ^2 + \left( \mathbb {E}^i_{\alpha b} (\varepsilon ^0_b - \varepsilon ^{\text {vp}, 0}_b) + \mathbb {E}^0_{\alpha c} (\varepsilon ^{\mathbb {E}, j}_c - \varepsilon ^{\text {vp}, \mathbb {E}, j}_c) \right) ^2 + \left( \mathbb {E}^0_{\alpha d} (\varepsilon ^{\text {Y}, k}_d - \varepsilon ^{\text {vp}, \text {Y}, k}_d)\right) ^2 \nonumber \\&\quad + \left( \mathbb {E}^i_{\alpha e} (\varepsilon ^0_e - \varepsilon ^{\text {vp}, 0}_e) + \mathbb {E}^0_{\alpha f} (\varepsilon ^{\mathbb {E}, j}_f - \varepsilon ^{\text {vp}, \mathbb {E}, j}_f) \right) \left( \mathbb {E}^0_{\alpha g} (\varepsilon ^{\text {Y}, k}_g - \varepsilon ^{\text {vp}, \text {Y}, k}_g) \right) \langle \xi ^{\mathbb {E}, k}, \xi ^{\text {Y}, k} \rangle . \end{aligned}$$
(36)

It may be remarked that \(\alpha \) denotes the individual component. Such, no summation is performed over this index.

4.3 Expectation and variance of the reaction force

The reaction force is an important global quantity and is computed by evaluating the internal forces

$$\begin{aligned} \varvec{F}= \int _{\Omega } \varvec{B}^T \cdot \varvec{\sigma }\,\textrm{d}V. \end{aligned}$$
(37)

at the nodes in a finite element framework with constrained degrees of freedom. In the equation above, \(\varvec{B}\) is the strain–displacement matrix, defined by \(\varvec{\varepsilon }= \varvec{B}\cdot \varvec{u}\).

Due to the direct link between reaction force and stress, expectation and variance result directly. The expectation of the reaction force is found as

$$\begin{aligned} \langle \varvec{F}\rangle = \int _{\Omega } \varvec{B}^T \cdot \mathbb {E}^0\cdot (\varvec{\varepsilon }^0 - \varvec{\varepsilon }^{\text {vp}, 0})\,\textrm{d}V. \end{aligned}$$
(38)

The variance is found as

$$\begin{aligned} \text {Var}(F_\alpha )&= \left( \int _\Omega B_{\alpha a} \left( \mathbb {E}^i_{a b} (\varepsilon ^0_b - \varepsilon ^{\text {vp}, 0}_b) + \mathbb {E}^0_{a c} (\varepsilon ^{\mathbb {E}, j}_c - \varepsilon ^{\text {vp}, \mathbb {E}, j}_c) \right) \ \textrm{d}V \right) ^2 \nonumber \\&\quad + \left( \int _\Omega B_{\alpha d} \left( \mathbb {E}^0_{d e} (\varepsilon ^{\text {Y}, k}_e - \varepsilon ^{\text {vp}, \text {Y}, k}_e)\right) \ \textrm{d}V \right) ^2 \nonumber \\&\quad + \left( \int _\Omega B_{\alpha f} \left( \mathbb {E}^i_{f g} (\varepsilon ^0_g - \varepsilon ^{\text {vp}, 0}_g) + \mathbb {E}^0_{f h} (\varepsilon ^{\mathbb {E}, j}_h - \varepsilon ^{\text {vp}, \mathbb {E}, j}_h) \right) \ \textrm{d}V \right) \nonumber \\&\quad \quad \left( \int _\Omega B_{\alpha i} \left( \mathbb {E}^0_{i j} (\varepsilon ^{\text {Y}, k}_j - \varepsilon ^{\text {vp}, \text {Y}, k}_j) \right) \ \textrm{d}V \right) \langle \xi ^{\mathbb {E}, k}, \xi ^{\text {Y}, k} \rangle \end{aligned}$$
(39)

for each component \(\alpha \). The summation is only carried out over the latin indices. As the reaction force is deterministically zero inside of the structure, the variance only has to be evaluated at the boundary with prescribed non-zero displacements.

5 Algorithmic implementation

For the algorithmic implementation of the material model, different approaches are known in literature, cf. [26]. Implicit schemes allow for large step sizes but tend to need a larger computational effort. For the comparison with the Monte Carlo method, a small computational effort needs to be preferred as otherwise the Monte Carlo method quickly becomes infeasible. Therefore, we perform a fully explicit time integration in order to keep the computational effort minimal. The timestep size is chosen accordingly in order to guarantee physically accurate results. Therefore, the update of internal variable and stress for the elasto-viscoplastic material model is given as

$$\begin{aligned} \varvec{\varepsilon }^{\text {vp}}_{n+1}&= \varvec{\varepsilon }^{\text {vp}}_{n} + \frac{\Delta t}{\eta }\left( ||\text {dev}(\varvec{\sigma }_n)|| - \sigma ^\text {Y} \right) _+ \frac{\text {dev}( \varvec{\sigma }_n)}{||\text {dev}(\varvec{\sigma }_n)||} \nonumber \\ \varvec{\sigma }_{n+1}&= \mathbb {E}\cdot (\varvec{\varepsilon }_{n+1} - \varvec{\varepsilon }^{\text {vp}}_{n+1}) \end{aligned}$$
(40)

for each timestep \(n+1\).

For the Monte Carlo method, the usual balance of linear momentum is solved at each timestep

$$\begin{aligned} \int _{\Omega } \delta \varvec{\varepsilon }\cdot \mathbb {E}\cdot (\varvec{\varepsilon }_{n+1} - \varvec{\varepsilon }^\text {vp}_{n+1}) \,\ \textrm{d}V - \int _\Omega \delta \varvec{u}_{n+1} \cdot \varvec{b}^* \,\ \textrm{d}V - \int _\Omega \delta \varvec{u}_{n+1} \cdot \varvec{t}^* \, \ \textrm{d}A = 0, \end{aligned}$$
(41)

each time with a different realization of the elasticity tensor \(\mathbb {E}\). Special attention is given to the fact that due to the modeling as a Gaussian random field unphysical negative material parameters could occur. In this case a resampling is done. Expectation and standard deviation are calculated in a post-processing step.

For the TSM, the scheme above is used for the calculation of the terms \(\varvec{\varepsilon }^0, \varvec{\varepsilon }^{\text {vp}, 0}\) with the elasticity tensor and yield limit being fixed to its expectation. For each Karhunen–Loéve term i of the elasticity tensor, an additional finite element calculation is performed for the calculation of \(\varvec{\varepsilon }^{\mathbb {E}, i}\) and \(\varvec{\varepsilon }^{\text {vp}, \mathbb {E}, i}\). Here, the balance equation

$$\begin{aligned} \int _\Omega \delta \varvec{\varepsilon }\cdot (\mathbb {E}^i \cdot \varvec{\varepsilon }_{n+1}^0 + \mathbb {E}^0 \cdot (\varvec{\varepsilon }_{n+1} - \varvec{\varepsilon }_{n+1}^{\text {vp}, \mathbb {E}, i})) \, \ \textrm{d}V = 0 \end{aligned}$$
(42)

is solved. We use an explicit scheme for the evolution of \(\varvec{\varepsilon }^{\text {vp}, \mathbb {E}, i}\), e.g., \(\varvec{\varepsilon }^{\text {vp}, \mathbb {E}, i}_{n+1} = f(\varvec{\varepsilon }^0_n, \varvec{\varepsilon }^{\text {vp}, 0}_n, \varvec{\varepsilon }^{\text {vp}, \mathbb {E}, i}_n)\) for Eq.  (22). Of course, alternatively, implicit or semi-implicit schemes can be used. The calculation of the terms \(\varvec{\varepsilon }^{\text {Y}, i}\) and \(\varvec{\varepsilon }^{\text {vp}, \text {Y}, i}\) linked to the stochastic yield limit require each an additional finite element calculation. This can be computationally prohibitive for large engineering simulations. Therefore, we propose a different approach. We make use of the Voigt bound for balance equation (19) by assuming constant strains. Consequently, we set \(\varvec{\varepsilon }^{\text {Y}, i}\) to zero. However, the evolution of \(\varvec{\varepsilon }^{\text {vp}, \text {Y}, i}\) should not be neglected. For this, we make use of an explicit evolution of \(\varvec{\varepsilon }^{\text {vp}, \text {Y}, i}\) for Eq.  (25). At each timestep, the expectation and variance of internal variable, stress, and reaction force are calculated following the equations as outlined in Sect. 4.

6 Numerical results

We investigate three different boundary value problems in order to showcase the models robustness, accuracy and efficiency. Specifically, we simulate a plate with a centered hole, a spring and a turbine blade.

The material parameters are chosen the same for all boundary value problems. The mean of the Young’s modulus is chosen as \({100} \text {MPa}\), the Poisson number as \(\nu = 0.3\). The Lamé parameter result according to the standard conversion. We assume a coefficient of variation of \(20\%\) of the Lamé parameters if not stated otherwise. The coefficient of variation is defined as the fraction of standard deviation to mean. Furthermore, we assume a full correlation between both Lamé parameters. The viscosity parameter is set as \({400} \text {GPas}\) and the expectation of the yield limit as \(\langle \sigma ^Y \rangle = {500} \text {MPa}\). For validation purposes, we compare against the standard Monte Carlo method (MC). Here, we choose 500 sampling points of the Monte Carlo method to balance computational effort with accuracy. Comparisons to results with a larger number of sampling points were made to make sure that the results can be treated as converged. A larger number of Monte Carlo sampling points would only shift the comparison more in favor of our method.

To quantify the accuracy of our method, we introduce two error measures. We compare the relative global error of the approximation of an field \(\varvec{a}\) against the Monte Carlo simulations as

$$\begin{aligned} \overline{\text {err}}(\varvec{a}) := & {} \frac{1}{\displaystyle \int _T \int _\Omega ||\varvec{a}^{\text {MC}}(t)|| \, \textrm{d}t \, \textrm{d}V} \int _T \int _\Omega ||\varvec{a}^{\text {MC}}(t) - \varvec{a}^{\text {TSM}}(t)|| \, \textrm{d}t \, \textrm{d}V. \end{aligned}$$
(43)

or equivalently in discretized form as

$$\begin{aligned} \text {err}(\varvec{a}) := & {} \frac{1}{\displaystyle \sum _{n=1}^{n_T} \sum _{e=1}^{n_\text {elem}} ||\varvec{a}^{e, \text {MC}}_n || \, \Omega ^e } \sum _{n=1}^{n_T} \sum _{e=1}^{n_\text {elem}} ||\varvec{a}^{e, \text {MC}}_n -\varvec{a}^{e, \text {TSM}}_n|| \, \Omega ^e. \end{aligned}$$
(44)

We also report on the relative error per timestep as

$$\begin{aligned} \text {err}^t(\varvec{a}, n) := & {} \frac{1}{\displaystyle \sum _{e=1}^{n_\text {elem}} ||\varvec{a}^{e, \text {MC}}_n || \, \Omega ^e } \sum _{e=1}^{n_\text {elem}} ||\varvec{a}^{e, \text {MC}}_n -\quad \varvec{a}^{e, \text {TSM}}_n|| \, \Omega ^e. \end{aligned}$$
(45)

As the Monte Carlo method only provides an estimate of the real answer, the reported error values should of course be taken with a grain of salt.

All simulations were performed on a consumer-grade laptop with an Intel i5-1145G7 processor and 16GB of RAM. The implementation was done in Julia [5] and the finite element framework Ferrite.jl [6].

6.1 Plate with hole

As the first boundary value problem, we investigate a plate with a hole. Only the upper-right quarter is discretized and symmetry boundary conditions are in place. The discretized part has a side length of \({10}\, \text {cm}\). A time-proportional displacement is applied on the top. In the first half of the simulation, tension occurs and afterward compression.

6.1.1 Homogeneous fluctuations

First, we assume homogeneous material fluctuations, i.e., a covariance length \(l \rightarrow \infty \) and therefore only one Karhunen–Loéve term. We set the coefficient of variation for all material parameters to 20%. The results for the spatial distribution of the expectation and the standard deviation of the stress at the end of the simulation for two different loading velocities are presented in Table 1. The maxima occur at the lower end of the plate near the cut out. As expected, the differences between the result from the TSM approach and the Monte Carlo method are minimal. The only differences are visible for the case of the lower loading velocity of \( {0.25}\, \text {mm/s}\). However, the TSM still provides a good estimate of the standard deviation. For the case of the loading velocity of \({0.5} \text {mm/s}\), we report the following error values: \(\text {err}(\langle \varvec{\sigma }\rangle ) = 1.4\%\), \(\text {err}(\text {Std}(\varvec{\sigma })) = 16.8\%\) and \(\text {err}(\langle \varvec{\sigma }\rangle \pm \text {Std}(\varvec{\sigma }))\) are 2.5% and 4.0%, respectively. The error values for the case of a lower loading velocity of \({0.25} \text {mm/s}\) are similar. Only the error of the standard deviation is significantly higher with \(\text {err}(\text {Std}(\varvec{\sigma })) = 26.4\%\). This is due to the fact that with a lower loading velocity the nonlinear viscoplastic strains are getting more significant. The most important observation is the computational time: the TSM needs less than a minute to arrive at results for expectation and standard deviation of all quantities whereas the Monte Carlo method with 500 sampling points needs nearly four hours. This in return means that the TSM only needs the same amount of time as 2.1 deterministic finite element simulations. We further present the reaction force for different loading speeds in Fig. 1. Corresponding to the loading speed, the total simulation time is adjusted such that the same maximal displacement is reached. The results of the TSM are presented in red and the results of MC are presented in black. The expectation is marked as a solid line, expectation ± standard deviation are shown as a dotted line. The similarity between the solutions over all loading speeds is striking. With deceasing loading velocity the viscous effects are more pronounced and the maximal force reached is reduced. It may be noted that for the lowest loading velocity the behavior shows similarity to perfect plasticity as the slope of the force tends to zero at \(t = {20}\, \text {s}\) and \(t = {40}\, \text {s}\). Nevertheless, the TSM is able to provide a near to perfect approximation. Also noticeable is the position of the point with zero standard deviation. It moves depending on the loading velocity showcasing the complex relationship of standard deviation and expectation.

Table 1 Comparison of approximation of the standard deviation of stress for two different loading velocities and computational time. The unit is Pascal
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Fig. 1
figure 1

Reaction force (expectation ± standard deviation) for different loading velocities. Black curves represent the reference Monte Carlo simulation results. Red lines represent the results obtained with the TSM

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6.1.2 Local fluctuations

We expand the investigation of the plate to problems with local material fluctuations. For this, we use the same boundary value problem but choose a correlation length of \({0.2}\, \text {m}\) resulting in 10 Karhunen–Loéve terms with an approximation error \(\text {err}^\rho \) of less than 5%. The same correlation length and therefore same number of Karhunen–Loéve terms is used for \(\lambda , \mu \) and \(\sigma ^\text {Y}\) but, of course, any modeling assumption is possible. We investigate two different options for the random material parameters. First, we assume that \(\lambda \) and \(\mu \) are stochastic and \(\sigma ^\text {Y}\) deterministic. Second, we investigate for the case of all three material parameters being stochastic. In particular, we investigate the choice of fully correlated material parameters. For both cases, we report the error of the expectation, standard deviation and expectation + standard deviation of stress and viscoplastic strains. These error measures are reported for a coefficient of variation of 10% and 20% of the material parameters in Table 2. The error over all timesteps for the expectation and expectation + standard deviation of the stress and viscoplastic strains are in the low single digit percentage range. In contrast, the error of the standard deviation is in the low two digit percentage range. This is not surprising as the standard deviation is indeed more complicated to approximate as the expectation. A larger ratio between standard deviation and expectation of the material parameters and more stochastic parameters naturally lead to a larger error value. The numbers can be better interpreted when the results for the spatial distribution are compared side by side. In Fig. 2, the spatial distribution of the standard deviation of the viscoplastic strains for stochastic \(\lambda \) and \(\mu \) and deterministic \(\sigma ^\text {Y}\) are compared during the unloading. The coefficient of variation of the stochastic material parameters were set to 20%. The results of TSM and MC are very close. The only visible difference is that the TSM underestimates the standard deviation of the internal variable slightly. However, for this time instance, an error of \(\text {err}^t \approx 10\%\) was reported. In our observation, error values up to 20% do correspond to a (visibly) good approximation. For error values lower than 5% no differences are visible. Anyway, our method comes with a much reduced computational effort. Instead of 500 MC simulations only \(\mathcal {O}(N)\) simulations are needed with N being the number of Karhunen–Loéve terms. Consequently, for low to mid dimensional problems the speed-up is at least one order of magnitude.

Table 2 Comparison of various error measures for local material fluctuations
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Fig. 2
figure 2

Comparison of the spatial distribution of the magnitude of the standard deviation of the viscoplastic strains for local material fluctuations

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6.2 Spring

As the second problem, we investigate a fully three-dimensional spring. It is discretized by approximately 2000 tetrahedral elements. A fixed boundary condition is applied on the lower end whereas a time-dependent displacement is applied on the top. In the first half of the simulation a time-proportional displacement is applied. After this, the displacement is hold constant. The material parameters \(\lambda \) and \(\mu \) are assumed stochastic with a coefficient of variation of 20%. The viscosity is set to \(\eta = {100}\, \text {GPas}\) and the yield limit to \(\sigma ^\text {Y} = {50}\, \text {MPa}\). The standard deviation of the stress is presented for the last timestep of the time-proportional loading in Fig. 3. At this point, the largest difference between TSM and reference solution can be expected as the viscoplastic strains have developed for a significant amount of time. Nevertheless, the approximation is surprisingly good, albeit the standard deviation is underestimated. Furthermore, we present the reaction force over time in Fig. 4. Most surprisingly is the near to zero standard deviation during the relaxation for \(t \ge {12}\, \text {s}\). The TSM approximates the increase and rapid decrease in the standard deviation well. However, around \(t \approx {10}\, \text {s}\) the expectation is (slightly) too high and the standard deviation too low. The difference in computation time is once again staggering. The TSM just needs the same time as 2.13 deterministic calculations.

Fig. 3
figure 3

Comparison of the spatial distribution of the magnitude of the standard deviation of the stress. All quantities are given in Pascal

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Fig. 4
figure 4

Reaction force (expectation ± standard deviation). Black curves represent the reference Monte Carlo simulation results. Red lines represent the results obtained with the TSM

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6.3 Turbine blade

The previous results encourage to use the TSM to investigate problems which were not feasible to simulate with Monte Carlo. Here, we investigate the stochastic behavior of a turbine blade. It represents an industry-scale problem as the turbine blade is discretized by 120000 elements. The viscosity is \(\eta = {100}\, \text {GPas}\) and the expectation of the yield limit is set to \(\langle \sigma ^\text {Y} \rangle = {100}\, \text {MPa}\). Homogeneous fluctuations are assumed and the coefficient of variation is set to 10% for all stochastic material parameters, i.e., Lamé parameters and yield limit. A time-proportional loading is applied on the top; the lower end is fixated. The results for the norm of expectation and standard deviation of both stress and viscoplastic strains at the end of the loading are presented in Table 3. A particularly interesting result is the position of the maxima of the standard deviation. These are occurring in the base of the turbine blade and around the mid of the blade. These are surprisingly enough positions with a particularly low value for the expectation of the stress. Similar effects have been observed before and are due to the nonlinearity introduced by the evolution law of the viscoplastic strains. The TSM simulation with 40 timesteps takes approximately 4h 45min. An investigation with the Monte Carlo method is simply not possible as well over a month of computation time would be needed.

Table 3 Results of TSM for spatial distribution of the magnitude of expectation and standard deviation for stress and viscoplastic strains
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7 Discussion

In this paper, the time-separated stochastic mechanics is presented as a robust method for uncertainty quantification for inelastic engineering simulations. It allows to approximate expectation and standard deviation of stress, internal variable and reaction force with a low number of deterministic FEM simulations. It is based on an expansion of all fields, i.e., strains and internal variable in a series around the expectation of the random variables. This allows to split the time-dependent from the stochastic behavior such that only one FEM simulation needs to be performed for each spatial variation of a material parameter. In this paper, the method is presented for the case of a elasto-viscoplastic material. Despite its low computational effort, the results are highly similar to reference Monte Carlo simulations for a variety of boundary value problems. Thus, the presented method enables new possibilities for the inclusion of uncertainty quantification in engineeringga simulations.