Phase-field modelling and analysis of rate-dependent fracture phenomena at finite deformation

Abstract

Fracture of materials with rate-dependent mechanical behaviour, e.g. polymers, is a highly complex process. For an adequate modelling, the coupling between rate-dependent stiffness, dissipative mechanisms present in the bulk material and crack driving force has to be accounted for in an appropriate manner. In addition, the resistance against crack propagation can depend on rate of deformation. In this contribution, an energetic phase-field model of rate-dependent fracture at finite deformation is presented. For the deformation of the bulk material, a formulation of finite viscoelasticity is adopted with strain energy densities of Ogden type assumed. The unified formulation allows to study different expressions for the fracture driving force. Furthermore, a possibly rate-dependent toughness is incorporated. The model is calibrated using experimental results from the literature for an elastomer and predictions are qualitatively and quantitatively validated against experimental data. Predictive capabilities of the model are studied for monotonic loads as well as creep fracture. Symmetrical and asymmetrical crack patterns are discussed and the influence of a dissipative fracture driving force contribution is analysed. It is shown that, different from ductile fracture of metals, such a driving force is not required for an adequate simulation of experimentally observable crack paths and is not favourable for the description of failure in viscoelastic rubbery polymers. Furthermore, the influence of a rate-dependent toughness is discussed by means of a numerical study. From a phenomenological point of view, it is demonstrated that rate-dependency of resistance against crack propagation can be an essential ingredient for the model when specific effects such as rate-dependent brittle-to-ductile transitions shall be described.

1 Introduction

The mechanical behaviour of many engineering materials depends on rate of deformation. For example, the response of polymers can be much more stiff or brittle when the loading rate is increased, see [1, 2]. The same applies for natural materials such as cheese [3] or confections [4]. In order to reduce experimental effort for design and testing of engineering products as well as for the optimisation of production processes of foods, the computational modelling and simulation of crack phenomena in rate-dependent materials is of increasing interest.

For the modelling of crack phenomena, the phase-field approach to fracture has become a well-established concept. Different from classical finite element approaches (FE), it enables to simulate crack growth without the need for remeshing. Furthermore, complex crack patterns that are not a priori known can be simulated in a straightforward manner, which especially makes the concept attractive compared to alternative approaches such as Cohesive zone elements [5] or the Extended-finite-element-method (X-FEM) [6]. The phase-field fracture approach goes back to the variational formulation of brittle fracture of Francfort and Marigo [7], who recast the Griffith criterion [8] for crack propagation into a variational setting. Bourdin et al. [9, 10] introduced a diffuse crack representation by means of the phase-field variable, which continuously varies from the intact to the fully broken material state. In other words, cracks are no longer seen as sharp discontinuities, but approximated over a finite length scale ${\ell _\text {c}}$. Making use of this smeared crack representation, a regularisation of the pseudo-energy functional is carried out. One key feature of this fundamental work of Bourdin is the $\Gamma$-convergence of the regularised functional against the sharp discontinuity model when the length scale ${\ell _\text {c}}$ tends to zero, i.e. ${\ell _\text {c}}\rightarrow 0$.

Based upon the diffuse crack representation introduced in [9], numerous models of fracture have been proposed. Not for all of these models, an underlaying pseudo-energy functional can be provided, from which the governing differential equations can be derived by means of variational arguments. Moreover, even if, $\Gamma$-convergence for ${\ell _\text {c}}\rightarrow 0$ is not necessarily preserved. Nevertheless, these models, which are based on the smeared description of crack topology, are typically referred to as phase-field models of fracture. Furthermore, although a lack of $\Gamma$-convergence is undesirable from a theoretical point of view, many of these models enable to take several possibly complex influencing factors into account, e.g. fatigue effects [11,12,13] or concentration of hydrostatic stress [14, 15], and have proven of value. Models of brittle fracture that include several advancements with respect to the fundamental work [9] have been proposed within both the infinitesimal strain regime [16,17,18,19,20] as well as for finite deformation [21,22,23]. Very recently, the phase-field approach to fracture also is combined with machine learning and data-driven approaches [24,25,26,27,28]. Furthermore, fracture phase-field modelling has been advanced towards elasto-plastic materials, see [29] for an overview on several approaches within the infinitesimal strain setting. For the performance of these ductile fracture models, the description of interaction between inelastic dissipative mechanisms and crack growth has revealed crucial. In particular, in the absence of an adequate coupling, crack patterns that are experimentally observed in metals, for instance, can not be reproduced, see e.g. [30, Fig. 14]. Different manners of introducing such a coupling are proposed, including non-energetic ductile fracture driving forces based on accumulated plastic strain [31, 32], and an enhanced degradation function which, in addition to the phase-field variable, depends on plastic deformation and results in a distinct plastic contribution to the fracture driving force [30, 33]. Furthermore, instead of a fracture driving force related to inelastic mechanisms, degradation of fracture toughness depending on equivalent plastic strain is introduced [34]. Several other phase-field models of ductile fracture are based upon a pseudo-energy functional in which both elastically stored energy and a plastic quantity, which is referred to as plastic work or plastic energy, are assumed to degrade upon fracture. Depending on the specific formulation, the plastic contribution to free energy actually corresponds to hardening terms [35, 36] or accumulated plastic dissipation [37,38,39].

More recently, the approach is combined with rate-dependent models for the deformation of the bulk material. A first phase-field fracture model for viscoelastic solids is proposed by Schänzel [32], where a non-energetic fracture driving force based on a generalised principal stress criterion is adopted. Alternative driving forces based on energetic or thermodynamic arguments are introduced by Shen et al. [40] as well as Liu et al. [41] within the kinematically linear regime and by Loew et al. [42, 43] within the linear viscoelasticity framework [44] at finite deformation. In these models, a viscous dissipative contribution is incorporated into the degraded free energy and thus enters fracture driving force. Different from the aforementioned models, only equilibrium and over-stress parts of the strain energy density are assumed to promote crack propagation by Yin and Kaliske [45], who combined the phase-field approach to fracture with a model of finite viscoelasticity [46]. Recently, similar formulations are adopted by Brighenti et al. [47] based on statistical mechanics-based equations for the response of the bulk material, and in [48] where the rate- and temperature-dependent behaviour of polymer nanocomposites is investigated. In some of these models based on either non-energetic or energetically motivated driving forces [32, 42], the viscosity assumed for the evolution of phase-field that originally is solely numerically motivated, cf. [17, 18], is understood as a material parameter and identified from experimental data. In the recent work of Dammaß et al. [49, 50], a unified energetic phase-field model for fracture of viscoelastic solids has been presented in the kinematically linear regime. Depending on the specific choice of the degradation functions and model parameters, respectively, the modelling approaches of [40, 42, 43] or [45, 47] are retained as limiting cases of the present model and by means of representative numerical studies, the coupling between viscous effects and fracture is analysed.

Compared to the rate-dependent behaviour of the bulk material, less efforts have been devoted to the study of strain rate-dependent resistance against fracture. Miehe et al. [31] suggested a phenomenological ansatz for the rate-dependent toughness in order to investigate the brittle-to-ductile fracture mode transition observed in the Kalthoff–Winkler experiment, i.e. for shear-loaded metals. Yin et al. [51] assumed the toughness of a linear elastic material to depend on rate of deformation. In their formulation, dissipation due to crack formation is incorporated into the free energy so that additional stress contributions are obtained from the rate-dependent fracture toughness. In these two models [31, 51], rate-independent models for the deformation of the bulk are considered. To the best of the authors’ knowledge, so far, there are no phase-field models that consider both a rate-dependent toughness and a model of rate-dependent deformation of the bulk.

Fig. 1

Modular structure and flexibility of the proposed model for rate-dependent fracture phenomena

In the present contribution, a thermodynamically consistent phase-field model of fracture of materials with rate-dependent behaviour is presented. For this purpose, the previously introduced pseudo-energy functional [50], which consists of the free energy that includes a contribution related to viscous dissipative mechanisms, and the fracture contribution is advanced towards the finite viscoelasticity setting of Reese and Govindjee [46]. Depending on the specific choice of the model parameters, the modelling approaches of [40, 42, 43] or [45, 47] can be retained as limiting cases. Based on experimental data for an Ethylene Propylene Diene Monomer (EPDM) rubber from the literature [42], the model parameters for the response of the bulk and the fracture behaviour are identified and model predictions are qualitatively and quantitatively verified on experimental results. In doing so, two assumptions for the fracture driving force, i.e. whether there shall be a contribution related to viscous dissipation or not, are investigated. With the aim of studying the possible influence of such a driving force component on the crack path, an asymmetrical setup is studied in addition to the symmetrical ones considered in recent publications, e.g. [42, 45]. Furthermore, based on experimental evidence on strain rate-dependent fracture toughness, cf. [3, 52, 53], and motivated from a phenomenological point of view, a rate-dependent resistance against fracture is introduced. A numerical study on the coupling between rate-dependent resistance against crack propagation and viscoelastic bulk response is then performed. An overview on the structure of the proposed unified model is given in Fig. 1.

The paper proceeds as follows. In Sect. 2, the proposed phase-field model of fracture in rate-dependent materials at finite deformation is presented and its thermodynamic consistency is proven. Subsequently, in Sect. 3, algorithmic aspects are addressed. In Sect. 4, the model parameterization is described and various numerical examples serve for validation and analysis of the model. A short summary and an outlook regarding the future work is given in Sect. 5. In the Appendix, information on the tangent for the local Newton iteration and the global material tangent is given.

Within this paper, italic symbols are used for scalar quantities (d, $\varPsi$) and bold italic symbols for vectors ($\varvec{u}$). For Second-order tensors, bold non-italic letters ($\textbf{T}$, $\varvec{\uptau }$) are used, whereas fourth-order tensors are written in Blackboard bold ($\mathbb {C}$).

2 Phase-field formulation

In this section, the phase-field model of fracture in materials with rate-dependent behaviour is presented. At first, the general energetic formulation of fracture in viscoelastic materials derived in [50] is extended to the finite deformation setting. Subsequently, the specific constitutive assumptions are outlined and the rate-dependent fracture toughness is introduced. Finally, governing equations are provided and thermodynamic consistency is proven.

Fig. 2

Diffuse representation of a crack within a domain that undergoes finite deformation

2.1 Pseudo-energy functional

Diffuse crack representation Motivated by the fundamental work of Bourdin [9], cracks are described in a diffuse manner by means of the fracture phase-field variable

$$\begin{aligned} d: \varOmega _0\times [0,t] \rightarrow [0,1] \text {,}\quad (\varvec{X}, t) \mapsto d(\varvec{X},t) \end{aligned}$$

(1)

which continuously varies from the intact ($d=0$) to the fully broken ($d=1$) material state. Figure 2 illustrates the concept of diffuse crack representation. Using this variable, following Miehe et al. [17], a functional

$$\begin{aligned} \gamma _{\ell _\text {c}}= \frac{1}{4 \, {\ell _\text {c}}} \left( d^2 + 4 \, {\ell ^2_\text {c}}\, \nabla _{\varvec{X}}d \cdot \nabla _{\varvec{X}}d \right) \end{aligned}$$

(2)

can be defined,¹ in which the length scale parameter ${\ell _\text {c}}$ controls the characteristic width of the diffuse crack. The Nabla operator with respect to the reference coordinate is given by

$$\begin{aligned} \nabla _{\varvec{X}}\, \circ = \sum _{K=1}^N \varvec{e}_K \, \left( \frac{\partial \, \circ }{\partial X_K}\right) \text {,}\end{aligned}$$

(3)

with $\varvec{e}_K$ denoting K-th basis vector of the Cartesian reference coordinate frame. This functional $\gamma _{\ell _\text {c}}$ can be understood as a crack surface density, which has been discussed in [16, 17, 31, 56] based on geometrical arguments. With this functional at hand, the density of rate of dissipation due to crack evolution is defined as

$$\begin{aligned} {\dot{\varPhi }}^\textrm{fr}= {\dot{\gamma }}_{\ell _\text {c}}\, {\mathcal {G}}_\text {c}\text {,} \end{aligned}$$

(4)

which refers to a volume element of the reference or undeformed configuration $\varOmega _0\subset \mathbb {R}^N$ of the N-dimensional domain under consideration. The parameter ${\mathcal {G}}_\text {c}>0$ quantifies the resistance of the material against fracture. This definition of ${\dot{\varPhi }}^\textrm{fr}$ is conceptionally similar to the hypothesis of Griffith [8] who assumed the increase of dissipation due to crack growth, which is referred to as surface energy, to be proportional to the increment of crack surface. As a consequence, in analogy to the proportionality constant of classical fracture mechanics, ${\mathcal {G}}_\text {c}$ is referred to as fracture toughness in this work, which is line with e.g. [34, 51, 57]. Nevertheless, it has to be noted that in the context of the proposed model and other approaches that do not guarantee $\Gamma$-convergence, ${\mathcal {G}}_\text {c}$ in general does not exactly correspond to the toughness or critical energy release rate according to the sharp crack description, which correlates with stress intensity factors of classical fracture mechanics. Instead, ${\mathcal {G}}_\text {c}$ rather has to be seen as a measure of resistance against fracture that is valid within the diffuse framework. Classically, ${\mathcal {G}}_\text {c}$ is assumed to be a constant, so that the density of dissipation due to crack evolution, which can be referred to as density of fracture-pseudo energy, can be written as

$$\begin{aligned} \varPhi ^\textrm{fr}= \gamma _{\ell _\text {c}}\, {\mathcal {G}}_\text {c}\text {.} \end{aligned}$$

(5)

However, ${\mathcal {G}}_\text {c}$ can explicitly depend on the position in space in heterogeneous materials [58, 59]. Furthermore, in the recent literature on phase-field modelling of fracture, ${\mathcal {G}}_\text {c}$ is assumed to change during fatigue life, see e.g. [12], or due to plastic deformation [34, 60]. In these cases, the fracture-pseudo energy can not be assumed to be proportional to $\gamma _{\ell _\text {c}}$. Instead, in order to determine $\varPhi ^\textrm{fr}$, the process history has to be accounted for. In the following, the phase-field framework is set up for which ${\mathcal {G}}_\text {c}=\text {const}.$ is assumed, first. Subsequently, the model is extended to account for a fracture toughness that depends on rate of deformation in Sect. 2.4.

Pseudo-energy functional For a domain $\varOmega _0$ affected by cracks that are represented by means of the phase-field variable d, the pseudo-energy functional

$$\begin{aligned} \varPi _{\ell _\text {c}}= \varPi ^\textrm{sd}_{\ell _\text {c}}+ \varPi ^\textrm{fr}_{\ell _\text {c}}= \int \limits _{ \varOmega _0} \varPsi + \varPhi ^\textrm{fr} \; \textrm{d} V \end{aligned}$$

(6)

can be defined,² which consists of the stored free energy $\varPi ^\textrm{sd}_{\ell _\text {c}}$, with its density with respect to the reference configuration denoted by $\varPsi$, and the fracture pseudo-energy $\varPi ^\textrm{fr}_{\ell _\text {c}}$ with its density given by $\varPhi ^\textrm{fr}$, which can be assumed to remain apparently stored within the material.

In the phase-field setting, the decrease of free energy due to fracture is expressed by means of the degradation function

$$\begin{aligned} g: [0,1] \rightarrow [0,1] \text {,}\quad d \mapsto g(d) \end{aligned}$$

(7)

which has to fulfil the conditions

$$\begin{aligned} \begin{aligned} g(d=0)=1 \text {,} &g(d=1)=0 \text {,}\\ \frac{\, \partial g}{\, \partial d} \le 0 \text {,} &\left. \frac{\, \partial g}{\, \partial d}\right| _{d=1}=0 \text {.} \end{aligned} \end{aligned}$$

(8)

By means of g(d), the degraded reference free energy density can be written as

$$\begin{aligned} \varPsi = g(d) \, \psi \text {,}\end{aligned}$$

(9)

with the virtually undamaged density of free energy, i.e. the amount of free energy that would be stored in the material in the absence of damage, denoted by $\psi$.

Generalisation for inelastic material response Following the previous work [50] and similar to phase-field fracture models for elasto-plastic materials, the free energy density

$$\begin{aligned} \varPsi = g_\text {st}(d)\, \psi ^\text {st}+ \beta _\text {vi}\, g_\text {vi}(d)\, \psi ^\text {vi}\; =: \varPsi ^\text {st}+ \varPsi ^\text {vi} \end{aligned}$$

(10)

is assumed to be additively decomposed into two essential ingredients.³ Naturally, the first one is the effectively stored strain energy $\varPsi ^\text {st}$. In addition, in order to adequately account for the coupling between inelastic deformation and fracture mechanisms, a free energy contribution $\varPsi ^\text {vi}$ related to accumulated viscous dissipation is assumed. Contributions similar to $\varPsi ^\text {vi}$ are also considered in other recent phase-field models of fracture in viscoelastic materials [40, 42]. Furthermore, analogue terms are widely spread in modelling of failure in elasto-plastic materials [36,37,38,39, 61], where a free energy contribution related to inelastic deformation, which is degraded in case of crack growth can be essential for the description of ductile fracture, cf. [29, 30].⁴ In $\varPsi ^\text {vi}$, in order to keep the formulation as general as possible, the parameter $\beta _\text {vi}\in [0,1]$ is introduced as a weight, cf. [39] and [40] or [37, 38] for analogous assumptions. This constant $\beta _\text {vi}$ is conceptionally similar to the Taylor–Quinney parameter widely spread in the modelling of plasticity [62], which quantifies the amount of plastic work that remains stored in the material. For a rigorous motivation and interpretation of $\varPsi ^\text {vi}$ from a physical point of view, and a numerical investigation in the kinematically linear regime, the reader is referred to [50].

The specific definitions of $\psi ^\text {st}$ and $\psi ^\text {vi}$ considered in this work are given in Sect. 2.2.⁵ For the two contributions to the free energy, any degradation functions $g_\text {st}$ and $g_\text {vi}$ satisfying the conditions (8) can be considered, which, in general, do not have to coincide. In the literature, different approaches have been taken, e.g. quartic and cubic expressions [63, 64], a sinusoidal ansatz [34, 45], and parametric functions that include additional parameters, which can be fitted to the behaviour of a specific material [20, 64,65,66]. Within the scope of this publication, $g_\text {st}(d)\equiv g_\text {vi}(d)\equiv g(d)$ is assumed. Furthermore, the frequently adopted [9, 17, 35, 40] quadratic function

$$\begin{aligned} g(d) = (1-k) \, (1-d)^2 + k \text {,} \end{aligned}$$

(11)

in which a small residual k is included in order to enhance numerical stability, is considered.

2.2 Viscoelastic bulk response

2.2.1 Kinematics

The displacement of a material point with the coordinate $\varvec{X} \in \varOmega _0$ in the reference configuration is denoted by

$$\begin{aligned} \varvec{u}(\varvec{X},t) = \varvec{\chi }(\varvec{X},t) - \varvec{X} \text {,}\end{aligned}$$

(12)

wherein

$$\begin{aligned} \varvec{\chi }(\varvec{X},t): \varOmega _0\times [0,t] \rightarrow \varOmega \text {,}\quad (\varvec{X}, t) \mapsto \varvec{x}(\varvec{X},t) \end{aligned}$$

(13)

is the motion function, which can be assumed to be bijective and continuous in space and time. The deformation gradient $\textbf{F}$ and its determinant J are then given by

$$\begin{aligned} \textbf{F}= \left( \nabla _{\varvec{X}}\varvec{\chi }\right) ^\top \qquad \text {and} \qquad J= \det \, \textbf{F}> 0 \text {.}\end{aligned}$$

(14)

For the rate-dependent deformation behaviour of the material, the approach of Reese and Govindjee [46] is pursued and a generalised Maxwell model is adopted as shown in Fig. 3.⁶

Fig. 3

Generalised Maxwell element—constitutive assumptions

In the non-equilibrium, or over-stress branch, deformation is assumed to consist of an elastic and an inelastic viscous portion, and the deformation gradient is multiplicatively decomposed into

$$\begin{aligned} \textbf{F}= {\textbf{F}^\textrm{el}}\cdot {\textbf{F}^\textrm{vi}}\text {,}\end{aligned}$$

(15)

accordingly. Furthermore, following Flory [67], a decomposition of the deformation gradient into volumetric and isochoric parts is applied. For the equilibrium branch, the split is given by

$$\begin{aligned} \textbf{F}= J^{1/3} \, {\textbf{I}}\cdot \overline{\textbf{F}}, \quad \overline{\textbf{F}}= J^{-1/3} \, \textbf{F}, \quad \det \, \overline{\textbf{F}}= 1 \text {,} \end{aligned}$$

(16)

wherein ${\textbf{I}}$ designates the second-order unit tensor and $\overline{\textbf{F}}$ is the isochoric portion of the deformation gradient. For the the non-equilibrium branch, ${\textbf{F}^\textrm{el}}$ and ${\textbf{F}^\textrm{vi}}$ are decomposed separately. Considering, for example, the elastic portion of deformation, its isochoric part is given by

$$\begin{aligned} {\overline{\textbf{F}}^\textrm{el}}= {{J^\textrm{el}}}^{-1/3} \, {\textbf{F}^\textrm{el}}\text {,}\qquad \det \, {\overline{\textbf{F}}^\textrm{el}}= 1 \text {,} \end{aligned}$$

(17)

wherein ${J^\textrm{el}}= \det \, {\textbf{F}^\textrm{el}}$. For the specific definition of the material model, the positive definite left and right Cauchy-Green deformation tensors, $\textbf{b}= \textbf{F}\cdot \textbf{F}^\top$ and $\textbf{C}= \textbf{F}^\top \cdot \textbf{F}$, as well as their elastic counterparts ${\textbf{b}}^\textrm{el}= {\textbf{F}^\textrm{el}}\cdot {{\textbf{F}^\textrm{el}}}^\top$ and ${\tilde{\textbf{C}}}^\textrm{el}= {{\textbf{F}^\textrm{el}}}^\top \cdot {{\textbf{F}^\textrm{el}}}$, are used, respectively. It has to be noted that ${\tilde{\textbf{C}}}^\textrm{el}$ does not refer to the reference configuration, but to a fictitious intermediate configuration defined by ${\textbf{F}^\textrm{vi}}$. The tilde symbol ${\tilde{\circ }}$ is introduced to mark quantities which refer to this inelastic intermediate configuration. Isotropy of the material is assumed and the constitutive equations are specified in terms of principal stretches $\lambda _\alpha$ and ${\lambda _{\beta }^\textrm{el}}$, which are obtained from the spectral decompositions

$$\begin{aligned}{} & {} \textbf{b}= \sum \limits _{\alpha =1}^{N_\lambda } \lambda _\alpha ^2 \, \textbf{p}_\alpha \qquad \text {or} \qquad \textbf{C}= \sum \limits _{\alpha =1}^{N_\lambda } \lambda _\alpha ^2 \, \textbf{P}_\alpha \text {,}\end{aligned}$$

(18)

$$\begin{aligned}{} & {} {\textbf{b}}^\textrm{el}= \sum \limits _{\beta =1}^{{N^\textrm{el}_\lambda }} {\lambda _{\beta }^\textrm{el}}^2\; {\textbf{p}_{\beta }^\textrm{el}}\qquad \text {or} \qquad {\tilde{\textbf{C}}}^\textrm{el}= \sum \limits _{\beta =1}^{{N^\textrm{el}_\lambda }} {\lambda _{\beta }^\textrm{el}}^2\; {\tilde{\textbf{P}}_{\beta }^\textrm{el}}\text {,}\end{aligned}$$

(19)

in which $N_\lambda \in \{1, 2, 3\}$ and ${N^\textrm{el}_\lambda }\in \{1, 2, 3 \}$ are the number of pair-wise different principal stretches $\lambda _\alpha$ and elastic principal stretches ${\lambda _{\beta }^\textrm{el}}$, respectively. The second-order projection tensors, or eigenvalue-base tensors, are obtained from

$$\begin{aligned} \textbf{p}_\alpha = \delta _{1 N_\lambda } \, \textbf{I}\, + \prod \limits _{\gamma } \frac{\textbf{b}- \lambda ^2_\gamma \, {\textbf{I}}}{\lambda ^2_\alpha - \lambda ^2_\gamma } \quad , \quad \gamma \in [1,N_\lambda ]\setminus \alpha \subset \mathbb {N}\text {,} \end{aligned}$$

(20)

with the Kronecker delta $\delta _{\varrho \sigma }$ given by

$$\begin{aligned} \delta _{\varrho \sigma } = \left\{ \begin{array}{c} 1, \quad \varrho = \sigma \\ 0, \quad \varrho \ne \sigma \\ \end{array} \right. \text {,} \end{aligned}$$

(21)

and equivalent relations for $\textbf{P}_\alpha$, ${\textbf{p}_{\beta }^\textrm{el}}$ and ${\tilde{\textbf{P}}_{\beta }^\textrm{el}}$ [68, 69].⁷

2.2.2 Specification of free energy densities

Strain energy The strain energy density is additively decomposed into an equilibrium and over-stress part. Accordingly, for the virtually undamaged quantity $\psi ^\text {st}$,

$$\begin{aligned} \psi ^\text {st}= \psi ^\text {st,eq}(\textbf{C}) + \psi ^\text {st,ov}(\textbf{C}, {\textbf{F}^\textrm{vi}}) \end{aligned}$$

(22)

is defined, wherein $\textbf{C}$ and ${\textbf{F}^\textrm{vi}}$ form the set of independent thermodynamic state variables considered here, in addition to the phase-field variable d. Each contribution splits further into a volumetric portion ${}^\textrm{vol}\psi ^\text {st,eq}$ and ${}^\textrm{vol}\psi ^\text {st,ov}$, and an isochoric part ${}^\textrm{iso}\psi ^\text {st,eq}$ and ${}^\textrm{iso}\psi ^\text {st,ov}$, respectively. A compressible Ogden model [70] is assumed for both the equilibrium and non-equilibrium branches and the respective strain energy density contributions are defined to

$$\begin{aligned} \psi ^\text {st,eq}&= {}^\textrm{vol}\psi ^\text {st,eq}+{}^\textrm{iso}\psi ^\text {st,eq}\nonumber \\&= \frac{\kappa ^\textrm{eq}}{4} \left( J^2 - 2 \ln J -1 \right) \nonumber \\&\quad + \sum _{p=1}^{N_\textrm{O}^\textrm{eq}} \frac{\mu ^\textrm{eq}_{p}}{\alpha ^\textrm{eq}_{p}} \, \left( \sum _{\varrho =1}^{N_\lambda } \nu _\varrho \, {\bar{\lambda }}_\varrho ^{\alpha ^\textrm{eq}_{p}} -3 \right) \text {,} \end{aligned}$$

(23)

$$\begin{aligned} \psi ^\text {st,ov}&= {}^\textrm{vol}\psi ^\text {st,ov}+{}^\textrm{iso}\psi ^\text {st,ov}\nonumber \\&= \frac{\kappa ^\textrm{ov}}{4} \left( {{J^\textrm{el}}}^2 - 2 \ln {J^\textrm{el}}-1 \right) \nonumber \\&\quad + \sum _{p=1}^{N_\textrm{O}^\textrm{ov}} \frac{\mu ^\textrm{ov}_{p}}{\alpha ^\textrm{ov}_{p}} \, \left( \sum _{\sigma =1}^{{N^\textrm{el}_\lambda }} {\nu }^\textrm{el}_{\sigma } \, \left( {{\bar{\lambda }}^\textrm{el}_{\gamma }}\right) ^{\alpha ^\textrm{ov}_{p}} -3 \right) \text {,} \end{aligned}$$

(24)

wherein ${\bar{\lambda }}_\varrho = J^{-1/3} \, \lambda _\varrho$ and ${{\bar{\lambda }}^\textrm{el}_{\sigma }} = {J^\textrm{el}}^{-1/3} \, {\lambda _{\sigma }^\textrm{el}}$ are the isochoric total and elastic principal stretches following from  (16) and (17). Their algebraic multiplicity is given by $\nu _\varrho \in \{1, 2, 3\}$ and ${\nu }^\textrm{el}_{\sigma } \in \{1, 2, 3\}$, respectively. Furthermore, the compression moduli are denoted by $\kappa ^\textrm{eq}>0$ and $\kappa ^\textrm{ov}>0$, and $N_\textrm{O}^\textrm{eq}$, $\alpha ^\textrm{eq}_{p}$, $\mu ^\textrm{eq}_{p}> 0$, as well as $N_\textrm{O}^\textrm{ov}$, $\alpha ^\textrm{ov}_{p}$, $\mu ^\textrm{ov}_{p}>0$ are parameters of the Ogden models. From these constants, the initial shear moduli and the according Poisson’s ratios can be defined. For example, for the equilibrium branch, they read

$$\begin{aligned} \mu ^\textrm{eq}= \frac{1}{2} \sum \limits _{p=1}^{N_\textrm{O}^\textrm{eq}} \mu ^\textrm{eq}_{p}\, \alpha ^\textrm{eq}_{p}\quad \text {and} \quad \nu ^\textrm{eq}= \frac{3 \, \kappa ^\textrm{eq}- 2 \, \mu ^\textrm{eq}_{p}}{2 \, (3 \, \kappa ^\textrm{eq}+ \mu ^\textrm{eq}_{p})} \text {,}\end{aligned}$$

(25)

and similar relations hold for the non-equilibrium branch.

Viscous contribution The degraded free energy contribution $\varPsi ^\text {vi}$ related to inelastic mechanisms is designed such that a certain portion of accumulated viscous dissipation can enter the phase-field fracture driving force. Before defining the respective virtually undamaged quantity $\psi ^\text {vi}$ in the finite viscoelasticity framework, the simple setting of a uniaxial deformation in the kinematically linear regime is considered for motivational purpose. Then, in the absence of damage, viscous dissipation in a material described by means of the generalised Maxwell model takes the form

$$\begin{aligned} {D}^\textrm{vi,1D}= \mathop {\int }_{ 0}^{t} \eta \, {\dot{\varepsilon }^\textrm{vi}} \, \, {\dot{\varepsilon }^\textrm{vi}} \; \textrm{d} {\bar{t}} \text {,}\end{aligned}$$

in which ${\dot{\varepsilon }^\textrm{vi}}$ is the rate of inelastic deformation and $\eta$ designates the viscosity of the material. In order to generalise ${D}^\textrm{vi,1D}$, the tensor

$$\begin{aligned} {\textbf{d}}^\textrm{vi}= - \frac{1}{2} \,\mathcal {L}\left[ {\textbf{b}}^\textrm{el}\right] \cdot {{\textbf{b}}^\textrm{el}}^{-1} \text {,} \end{aligned}$$

(26)

is introduced as a measure of the rate of inelastic deformation in the finite viscoelasticity setting, wherein

$$\begin{aligned} \mathcal {L}\left[ {\textbf{b}}^\textrm{el}\right] = \textbf{F}\cdot \dot{\left( {{\textbf{C}}^\textrm{vi}}^{-1} \right) } \cdot \textbf{F}^\top \quad \text {with} \quad {{\textbf{C}}^\textrm{vi}}= {\textbf{F}^\textrm{vi}}^\top \cdot {\textbf{F}^\textrm{vi}} \end{aligned}$$

(27)

is the Lie derivative of ${\textbf{b}}^\textrm{el}$. Furthermore, a fully symmetric, positive definite, isotropic fourth-order tensor

(28)

is defined, in which $\mathbb {I}^\text {D}$,

$$\begin{aligned} \mathbb {I}^\text {D}_{klmn} = \frac{1}{2} \left( \delta _{km} \delta _{ln}+\delta _{kn} \delta _{lm} \right) - \frac{1}{3}\delta _{kl} \delta _{mn} \text {,}\end{aligned}$$

(29)

is the fully symmetric fourth-order deviator projection tensor. Therein, are viscosities with respect to the isochoric and volumetric portion of deformation, respectively. The virtually undamaged free energy density contribution related to viscous mechanisms is then defined to

$$\begin{aligned} \psi ^\text {vi}= \mathop {\int }_{ 0}^{t} {\textbf{d}}^\textrm{vi}\, :\mathbb {V}\, :{\textbf{d}}^\textrm{vi} \; \textrm{d} {\bar{t}} \text {,}\end{aligned}$$

(30)

which is positive and monotonically increasing in time.

Remark on the measure of rate of inelastic deformation The definition of ${\textbf{d}}^\textrm{vi}$ (26) can be written in an alternative form, which may be more intuitive. For this purpose, the inelastic velocity gradient

$$\begin{aligned} {\tilde{\textbf{l}}{}^\textrm{vi}}= {\dot{\textbf{F}}^\textrm{vi}}\cdot {\textbf{F}^\textrm{vi}}^{-1} \end{aligned}$$

(31)

is introduced. It refers to the intermediate configuration defined by ${\textbf{F}^\textrm{vi}}$. The counterpart of ${\tilde{\textbf{l}}{}^\textrm{vi}}$ transformed to the current configuration reads

$$\begin{aligned} {\textbf{l}}^\textrm{vi}= {\textbf{F}^\textrm{el}}\cdot {\tilde{\textbf{l}}{}^\textrm{vi}}\cdot {\textbf{F}^\textrm{el}}^{-1} \text {.} \end{aligned}$$

(32)

Assuming that there is no inelastic spin, i.e. ${\tilde{\textbf{l}}{}^\textrm{vi}}= {\tilde{\textbf{d}}^\textrm{vi}}$ with ${\tilde{\textbf{d}}^\textrm{vi}}= {{\,\textrm{sym}\,}}{\tilde{\textbf{l}}{}^\textrm{vi}}$ denoting the rate of inelastic deformation with respect to the viscous intermediate configuration, (27)₁ can be rewritten as

$$\begin{aligned} \mathcal {L}\left[ {\textbf{b}}^\textrm{el}\right] = - 2 \, {\textbf{F}^\textrm{el}}\cdot {\tilde{\textbf{d}}^\textrm{vi}}\cdot {\textbf{F}^\textrm{el}}^\top \end{aligned}$$

(33)

and

$$\begin{aligned} {\textbf{F}^\textrm{el}}\cdot {\tilde{\textbf{d}}^\textrm{vi}}\cdot {\textbf{F}^\textrm{el}}^{-1} = - \frac{1}{2} \,\mathcal {L}\left[ {\textbf{b}}^\textrm{el}\right] \cdot {{\textbf{b}}^\textrm{el}}^{-1} ={\textbf{d}}^\textrm{vi}\end{aligned}$$

(34)

holds, from which, together with the transformation rule (32), the definition of ${\textbf{d}}^\textrm{vi}$ as an Eulerian measure of rate of inelastic deformation becomes clear. For more details, the reader is referred to [71], where similar kinematic relations are derived in the context of plasticity.

2.3 Evolution of phase-field

The equation governing the evolution of the fracture phase-field variable is deduced from the pseudo-energy functional $\varPi _{\ell _\text {c}}$ by means of the variational derivative

(35)

wherein ${\eta _\text {f}}$ is introduced as a kinetic fracture parameter in order to avoid discontinuity of the field variables in time and for numerical purposes, i.e. for enhancing the stability of the solution scheme, cf. [35, 72] and $\varvec{N}$ denotes the outward-pointing unit normal vector on $\partial \varOmega _0$. For the simulations presented in Sect. 4, ${\eta _\text {f}}$ is chosen such small that its influence on the simulation results vanishes which is verified by means of a comparative study of different values.⁸

Inserting the definitions made in the previous sections into $\varPi _{\ell _\text {c}}$, the evolution equation (35)₁ takes the form

$$\begin{aligned} \begin{aligned} - {\eta _\text {f}}\, {\dot{d}}&= \frac{\partial g}{\partial d} \left( \psi ^\text {st}+ \beta _\text {vi}\, \psi ^\text {vi}\right) \\&\quad + {\mathcal {G}}_\text {c}\left( \frac{1}{2 \, {\ell _\text {c}}} d - 2\, {\ell _\text {c}}\, \nabla _{\varvec{X}}\cdot \nabla _{\varvec{X}}d \right) \end{aligned} \end{aligned}$$

(36)

from which it becomes clear that, depending on the specific choice of $\beta _\text {vi}$, fracture is driven by stored strain energy and the free energy contribution related to a portion of accumulated viscous dissipation. It has to be noted that in the present form (36), the evolution equation enables the phase-field variable to decrease, i.e. crack healing is not prohibited. Therefore, a modification is adopted which overcomes this issue, see Sect. 3.1.

2.4 Rate-dependent fracture toughness

For various materials, in addition to or instead of the deformation behaviour of the bulk material, the resistance against fracture has been reported to depend on rate of deformation. For instance, in elastomers, at low rates of deformation, chain entanglements can be resolved, which is not the case at high rates of deformation. Therefore, the number of chemical bonds that are broken when a crack propagates can be assumed to rise with rate of deformation and the fracture toughness increases accordingly, cf. [2, 73] for a more detailed discussion and experimental results. Furthermore, for several natural materials and foods, where the underlaying microscopic mechanisms can be more complex, a rate-dependency of ${\mathcal {G}}_\text {c}$ has been reported [3, 52, 53].⁹ Therefore, as an extension of the phase-field equation (36) that has been derived for ${\mathcal {G}}_\text {c}=\text {const}.$, the fracture toughness is considered to depend on deformation rate in what follows, which enables a maximum of flexibility in modelling rate-dependent fracture processes. For this purpose,

(37)

is introduced as a scalar measure of the rate of deformation $\textbf{d}$, which is given by

(38)

Furthermore, in line with [74], the sigmoid-shaped function

(39)

is adopted, here, see Fig. 4. The phase-field evolution equation is than, similar to the suggestions made in [32, 75], for instance, extended in non-variational manner. For the case of ${\mathcal {G}}_\text {c}(\textbf{d})$, it is rewritten to¹⁰

(40)

In Sect. 4.3, for different parameters ${\mathcal {G}}_\text {c}^1$, ${\mathcal {G}}_\text {c}^2$, c, $r_\text {ref}$, coupling between rate-dependent deformation and toughness is analysed.

Fracture pseudo-energy and rate-dependent toughness If fracture toughness is a function of rate of deformation and thus implicitly depends on time, density of fracture pseudo-energy $\varPhi ^\textrm{fr}$ no longer is proportional to the crack surface density functional $\gamma _{\ell _\text {c}}$ as it is the case for ${\mathcal {G}}_\text {c}=\text {const}.$ Instead, $\varPhi ^\textrm{fr}$ has to be determined by explicitly integrating ${\dot{\varPhi }}^\textrm{fr}$ according to (4) over the process history, i.e.

(41)

In addition, it is noted that, different from e.g. [51], in the proposed model, rate-dependency of fracture toughness does not affect the density of free energy $\varPsi$, since dissipation due to crack growth is not supposed to enter $\varPsi$. Accordingly, no additional stress terms arise from rate-dependent toughness, see the evaluation of the dissipation inequality below in Sect. 2.5.

Fig. 4

Rate-dependent fracture toughness function ${\mathcal {G}}_\text {c}\left( r(\textbf{d})\right)$

2.5 Stress tensor, viscous evolution and thermodynamic consistency

Under isothermal conditions, the second law of thermodynamics can be stated by means of the density of dissipation power $\dot{D}$ as

$$\begin{aligned} \dot{D}= \frac{1}{2} \, \textbf{T}\, :\dot{\textbf{C}} - {\dot{\varPsi }} \ge 0 \text {,}\end{aligned}$$

(42)

cf. [76], with $\textbf{T}$ denoting the second Piola–Kirchhoff stress tensor. For $\varPsi = \varPsi (\textbf{C}, {\textbf{F}^\textrm{vi}}, d)$, this inequality can be rewritten to

$$\begin{aligned} \dot{D}&= \left( \frac{1}{2} \, \textbf{T}- g(d) \, \frac{\partial \psi ^\text {st}}{\partial \textbf{C}} \right) \, :\dot{\textbf{C}} \underbrace{- \frac{\partial g}{\partial d} \left( \psi ^\text {st}+ \beta _\text {vi}\psi ^\text {vi}\, \right) \dot{d}}_{{\dot{D}}^\textrm{fr}} \nonumber \\&\quad \underbrace{- g(d) \left( \frac{\partial \psi ^\text {st}}{\partial {\textbf{F}^\textrm{vi}}} \, :\dot{{\textbf{F}^\textrm{vi}}} + \beta _\text {vi}\, {\textbf{d}}^\textrm{vi}\, :\mathbb {V}\, :{\textbf{d}}^\textrm{vi}\right) }_{{\dot{D}}^\textrm{vi}} \ge 0 \text {,}\end{aligned}$$

(43)

wherein the contributions to dissipation power density due to fracture, ${\dot{D}}^\textrm{fr}$, and viscous effects, ${\dot{D}}^\textrm{vi}$, can be identified. The standard argument that $\dot{D}\ge 0$ shall hold for arbitrary processes leads to the definition of stress

(44)

with the virtually undamaged equilibrium and over-stress tensors denoted by and , respectively, and the residual inequalities

(45)

Stress tensor Inserting the definitions of $\psi ^\text {st,eq}$ and $\psi ^\text {st,ov}$, Eqs. (23) and (24), into (44), the contributions to the second Piola–Kirchhoff stress tensor take the form

(46)

Table 1 Governing equations for the present model following the total Lagrangian approach: mechanical equilibrium (a), phase-field equation (b), viscous evolution (c), rate-dependent fracture toughness (d). Without loss of generality, volume forces are neglected in (a)

(47)

Residual inequalities As both $\psi ^\text {st}$ and $\psi ^\text {vi}$ are positive, $\beta _\text {vi}\in [0,1]$, and due to (8)$_3$, the condition ${\dot{D}}^\textrm{fr}\ge 0$ reduces to $\dot{d} \ge 0$, i.e. irreversibility of fracture. The fulfilment of this demand will be addressed in Sect. 3.1.

Due to (7), ${\dot{D}}^\textrm{vi}\ge 0$ reduces to

$$\begin{aligned} \frac{\partial \psi ^\text {st}}{\partial {\textbf{F}^\textrm{vi}}} \, :\dot{{\textbf{F}^\textrm{vi}}} + \beta _\text {vi}\, {\textbf{d}}^\textrm{vi}\, :\mathbb {V}\, :{\textbf{d}}^\textrm{vi}\le 0 \text {.} \end{aligned}$$

(48)

Making use of the relations outlined in Sect. 2.2, after some lengthy manipulations, the first term can be rewritten as

(49)

wherein

(50)

is the virtually undamaged Kirchhoff over-stress. For a more detailed derivation see also [46]. Then (48) takes the form

(51)

from which, in line with [46], the equation governing viscous evolution

(52)

is defined. By reason of $\beta _\text {vi}\in [0,1]$, the quadratic form obtained from inserting (52) into (51) is compatible with the second law of thermodynamics.

3 Algorithmic aspects

3.1 Irreversibility of fracture

In order to guarantee the irreversibility of fracture, the history variable approach of Miehe et al. [17] is pursued. For this purpose, the phase-field equation (40) is rewritten to

$$\begin{aligned} - \frac{{\eta _\text {f}}}{{\mathcal {G}}_\text {c}} \, {\dot{d}}&= \frac{\partial g}{\partial d} \, {\mathcal {H}} + \left( \frac{1}{2 \, {\ell _\text {c}}} d - 2\, {\ell _\text {c}}\, \nabla _{\varvec{X}}\cdot \nabla _{\varvec{X}}d \right) \text {,} \end{aligned}$$

(53)

wherein the history variable

$$\begin{aligned} {\mathcal {H}}= \max _{\tau \in [0,t]} \left\{ \frac{1}{{\mathcal {G}}_\text {c}(\textbf{d}(\tau ))} \, \left[ \psi ^\text {st}(\tau ) + \beta _\text {vi}\, \psi ^\text {vi}(\tau ) \right] \right\} \end{aligned}$$

(54)

comprises the maximum of fracture driving force which has occurred. By means of ${\mathcal {H}}$, crack healing effects are prevented.¹¹ If such a history variable was not considered, these unphysical phenomena could arise from either an increase in fracture toughness ${\mathcal {G}}_\text {c}(\textbf{d})$ or a decrease in virtually undamaged free energy $\psi$. With the modified form of the phase-field evolution (53) at hand, the governing equations of the model are summarised in Table 1 considering the total Lagrangian approach.

As an alternative to the history variable approach, in line with [18, 77], Dirichlet boundary conditions can be applied to the phase-field on all nodes

$$\begin{aligned} \varvec{X}_\textrm{irrBC}\in \left\{ \varvec{X} \in \varOmega _0\; \left| \; \exists \, \tau \in [0,t]: d(\varvec{X},\tau ) \ge d_\textrm{crit} \right. \right\} \end{aligned}$$

(55)

where the phase-field variable has reached a critical value $d_\textrm{crit}$:

$$\begin{aligned} d(\varvec{X}_\textrm{irrBC}) {\mathop {=}\limits ^{!}}1 \; \forall \; \varvec{X}_\textrm{irrBC}\text {.} \end{aligned}$$

(56)

For the setups analysed in Sect. 4, the two strategies have been compared, exemplary, and no relevant differences could be noticed.

3.2 Viscous evolution

For the integration of viscous evolution equation (52), an operator split scheme of predictor–corrector type is adopted as proposed in [46]. Within the scope of this well-established approach, the evolution of elastic deformation in the over-stress branch of the generalised Maxwell element

$$\begin{aligned} \dot{\textbf{b}}^\textrm{el} = \underbrace{\textbf{l}\cdot {\textbf{b}}^\textrm{el}+ {\textbf{b}}^\textrm{el}\cdot \textbf{l}^\top }_\textrm{predictor} + \underbrace{\textbf{F}\cdot \dot{\left( {{\textbf{C}}^\textrm{vi}}^{-1} \right) } \cdot \textbf{F}^\top }_\textrm{corrector} \text {,} \end{aligned}$$

(57)

is split into the contributions from change in total deformation, which is considered in the predictor step, and viscous evolution, which is accounted for in the inelastic corrector. For the predictor step, viscous deformation ${\textbf{F}^\textrm{vi}}$ or ${{\textbf{C}}^\textrm{vi}}$ is frozen, giving a trial state of elastic deformation at time step $t_n$ to

$$\begin{aligned} {}_n{\textbf{b}}^\textrm{el}_\textrm{tr}= {}_n\textbf{F}\cdot {}_{n-1}{{\textbf{C}}^\textrm{vi}}^{-1} \cdot {}_n\textbf{F}^\top \text {.}\end{aligned}$$

(58)

Subsequently, within the corrector step, (57) is evaluated for the total deformation assumed to be constant, i.e. $\textbf{l}= \textbf{0}$, which, with evolution equation (52) and kinematic relations (26) and (27) can then be written as

(59)

Due to isotropy, the principal directions of ${\textbf{b}}^\textrm{el}$, ${\textbf{b}}^\textrm{el}_\textrm{tr}$ and coincide, which makes the evaluation of (59) in terms of elastic principal stretches ${\lambda _{\beta }^\textrm{el}}$ attractive. For the viscosity tensor $\mathbb {V}$ defined according to (28), this leads to

(60)

wherein denote the principal components of the over stress deviator . Within the scope of the FE framework, differential equation (60) is integrated in an approximate manner by means of an exponential mapping ansatz and rewritten in terms of logarithmic elastic principal stretches ${\varepsilon _{\beta }^\textrm{el}}= \ln {\lambda _{\beta }^\textrm{el}}$ as

(61)

Generally, ${\varepsilon _{\beta }^\textrm{el}}$ are determined from an iterative solution of the system of non-linear algebraic equations $r_\beta = 0$ with $\beta \in [1,N] \subset {\mathbb {N}}$. However, in case of two-dimensional plane stress setups as considered in Sect. 4, in addition to these three equations, it has to be ensured that the out of plane stresses vanishes, i.e. must hold. In these cases, in addition to ${\varepsilon _{\beta }^\textrm{el}}$, the out of plane stretch $\lambda _3$ has to be determined from the system of equations

(62)

Regardless of whether a plane stress state is considered or not, the respective system of equations (61) or (62) is solved by means of a local Newton iteration scheme at each quadrature point. In the following, the procedure is briefly described for the case that a plane stress state has to be guaranteed. With the vector of unknowns then written as

$$\begin{aligned} x_\sigma := \left[ \lambda _3 \quad {\varepsilon _{1}^\textrm{el}} \quad {\varepsilon _{2}^\textrm{el}} \quad {\varepsilon _{3}^\textrm{el}} \right] ^\top \end{aligned}$$

(63)

and the local tangent matrix

$$\begin{aligned} K_{\varrho \sigma } = \frac{\partial w_\varrho }{\partial x_\sigma } \text {,}\end{aligned}$$

(64)

the linearisation of (62) around ${}^j x_\sigma$ is given as

(65)

For the specification of the derivatives $\partial {w_\varrho }/\partial {x_\sigma }$, the reader is referred to Appendix A. Based on the linearisation, the Newton procedure is carried out as summarised in Algorithm box 1. For this, at each increment $t_n$, the iteration scheme is initialised by means of

(66)

with

(67)

Within the iterative solution procedure, special attention has to be paid to $_\textrm{tr}{\varepsilon _{3}^\textrm{el}}$ as it needs to be updated after each local iteration j according to

(68)

due to the change of $\varepsilon _3 = \ln \lambda _3$.

3.3 Weak forms of the governing equations

For the derivation of the weak forms of mechanical equilibrium and phase-field equation, the test function spaces

$$\begin{aligned} \mathbb {W}_{u_j}&:= \left\{ \delta u_{j}\in {\mathbb {H}}^1(\varOmega _0) \, \left| \, \delta u_{j}= 0 \; \forall \, \varvec{X} \in \partial \varOmega _{0 \, u_j}\right. \right\} \text {,}\nonumber \\&\quad \quad j \in [1,N] \subset {\mathbb {N}} \text {,}\end{aligned}$$

(69)

and

$$\begin{aligned} \mathbb {W}_c= {\mathbb {H}}^1(\varOmega _0) \end{aligned}$$

(70)

are defined. Therein, ${\mathbb {H}}^1(\varOmega _0)$ is the Sobolev space of square integrable functions possessing square integrable derivatives in $\varOmega _0$, and $\partial \varOmega _{0 \, u_j}$ denotes the parts of the boundary where the j-component of the displacement vector $\varvec{u}$ is prescribed. Then, (a) and (b) from Table 1 are multiplied by

$$\begin{aligned} \delta \varvec{u}= \left[ \delta u_{1} \, \cdots \, \delta u_{N} \right] ^\top , \quad \delta u_{j}\in \mathbb {W}_{u_j}\text {,}\end{aligned}$$

(71)

and $\delta c\in \mathbb {W}_c$, respectively. Integration by parts and making use of the divergence theorem yields

$$\begin{aligned} \int \limits _{ \varOmega _0} \left( \textbf{T}\cdot \textbf{F}^\top \right) \, :\left( \nabla _{\varvec{X}}\delta \varvec{u}\right) ^\top \; \textrm{d} V - \int \limits _{ \partial \varOmega _0} \hat{\varvec{p}} \, \delta \varvec{u} \; \textrm{d} A = 0 \text {,} \end{aligned}$$

(72)

wherein $\hat{\varvec{p}}$ denotes the Piola traction vector with its components ${\hat{p}}_j$ prescribed on $\partial \varOmega _0{\setminus } \partial \varOmega _{0 \, u_j}$, and

$$\begin{aligned}&\int \limits _{ \varOmega _0} \left( \frac{\partial g}{\partial d} \, {\mathcal {H}} + \frac{1}{2 \, {\ell _\text {c}}} \, d + \frac{{\eta _\text {f}}}{{\mathcal {G}}_\text {c}} \, \dot{d} \right) \, \delta c+ 2 \, {\ell _\text {c}}\, \nabla _{\varvec{X}}d \cdot \nabla _{\varvec{X}}\delta c \; \textrm{d} V \nonumber \\&\quad + \int \limits _{ \partial \varOmega _0} \, 2 \, {\ell _\text {c}}\, \underbrace{\nabla _{\varvec{X}}d \cdot \varvec{N}}_{= \, 0,\text { cf. (35)}} \, \delta c \; \textrm{d} A = 0 \text {.} \end{aligned}$$

(73)

Time discrete forms are obtained by approximating the respective rates using an Euler backward scheme. For spatial discretization, Galerkin’s method is applied. Then, the discretized equations are implemented into a standard finite element framework. The coupled problem is solved by means of a staggered approach, i.e. the mechanical equilibrium and the phase-field equation are solved in sequential manner. Multiple iterations of this decoupled solution scheme are performed until a staggered convergence criterion is fulfilled, which is based on the update of the nodal degrees of freedom. Furthermore, adaptive control of the time step size is employed based on a heuristic scheme. For this, depending on the number of global Newton iterations as well as on the number of staggered loops, the time step is either lowered or increased. In addition, if the convergence criteria are not met after a certain number of Newton iterations or staggered loops, respectively, a cut back to the last converged increment is performed and the time step size is lowered, accordingly. Information on the material tangent that is required for the iterative solution of (72) is given in Appendix B.

4 Representative simulations

In this section, several numerical examples are presented in order to analyse the characteristics of the present model and to demonstrate its flexibility in describing different responses. Furthermore, the comparison of numerical predictions to experimental results of Loew et al. [42] serves for validation of its predictive capabilities.

Fig. 5

Stress response of the bulk material under homogeneous uniaxial tension—experimental data [42] versus present model for three different stretch rates ${\dot{\lambda }}$

4.1 Parameter identification from experimental data

Within this publication, the viscoelastic behaviour of EPDM rubber is considered that has been experimentally analysed in [42].

Bulk response At first, the parameters describing the deformation of the bulk material are determined. For this purpose, the averaged stress-stretch curves from [42, Fig. 6] are considered as depicted in Fig. 5. For three different rates of deformation, these curves have been identified from uniaxial tension tests with dumbbell specimens. A homogeneous uniaxial stress state is assumed and damage is not taken into account, here. Furthermore, as no information on deformation in transversal direction is available, $\nu ^\textrm{eq}=\nu ^\textrm{ov}=0.48$ is set in order to account for the high resistance against volumetric deformation that is typically observed for rubber. For isochoric and volumetric deformation, an identical relaxation time

(74)

is assumed. The Ogden parameters $\mu ^\textrm{eq}_{p}, \alpha ^\textrm{eq}_{p}, \mu ^\textrm{ov}_{p}, \alpha ^\textrm{ov}_{p}$ as well as $\tau$ are then identified by means of minimising the deviation between experimental data and model prediction. In doing so, following [70, p. 305], it is demanded that the constants satisfy the requirements

$$\begin{aligned} \alpha ^\textrm{eq}_{p}\, \mu ^\textrm{eq}_{p}\ge 0 \text {and} \alpha ^\textrm{eq}_{p}\in (-\infty ,-1) \cup (2,\infty ) \end{aligned}$$

(75)

for any p and similar constraints for the non-equilibrium branch. To this end, in Matlab R2020b, the GlobalSearch strategy together with the fmincon algorithm for constrained optimisation problems is employed.¹² For an adequate approximation of the material behaviour, two Ogden exponents have revealed necessary for both the equilibrium and over-stress branch, respectively, i.e. $N_\textrm{O}^\textrm{eq}=N_\textrm{O}^\textrm{ov}=2$.¹³ The parameters obtained are summarised in Table 2. From Fig. 5 it becomes clear that the finite viscoelasticity formulation together with the Ogden approach allows for a very good approximation of the experimental results over the entire range of stretch $\lambda \in [1,2.5]$ that has been experimentally investigated. Furthermore, the present model enables to capture the rate-dependent response in a more reliable manner then the linear viscoelasticity model based on the Yeoh-type strain energy density [42].¹⁴

Table 2 Parameters of the finite viscoelasticity model for the deformation of the bulk material

Identification of the fracture parameters With the calibrated bulk deformation model at hand, the fracture phase-field is parameterized from SENT experiments, i.e. specimens with a single pre-existing notch under tension. These experiments have been conducted at two rates of prescribed displacement [42]. The according specimen geometry is depicted in Fig. 6.

Fig. 6

SENT—setup considered for the identification of ${\mathcal {G}}_\text {c}$

For the numerically motivated kinetic fracture parameter and the residual stiffness, the values ${\eta _\text {f}}=10^{-4}\,{\hbox {Ns/mm}^{2}}$ and $k=10^{-10}$, respectively, are chosen. In a convergence study, these values have revealed sufficiently small so that the influence of ${\eta _\text {f}}$ and k on the simulation results vanishes. For all the simulations presented in this paper, the pre-existing cuts within the specimens are modelled geometrically by means of notches.¹⁵

Fig. 7

SENT—experimental data [42] versus model for approaches A ($\beta _\text {vi}=0$) and B ($\beta _\text {vi}=1$)

Table 3 Parameters of the phase-field model calibrated for EPDM rubber with ${\mathcal {G}}_\text {c}= \mathrm {const.}$ assumed

Fig. 8

SENT—evolution of the energetic quantities for $\dot{\bar{u}} = {25}\,{\hbox {mm} \, \hbox {min}^{-1}}$ and the model approaches A ($\beta _\text {vi}=0$) and B ($\beta _\text {vi}=1$). Work of external forces $W^\textrm{ext}$ (grey), pseudo-energy functional $\varPi _{{\ell _\text {c}}} = \varPi ^\textrm{sd}_{\ell _\text {c}}+ \varPi ^\textrm{fr}_{\ell _\text {c}}$ (blue), stored free energy $\varPi ^\textrm{sd}_{\ell _\text {c}}$ (orange), fracture pseudo-energy $\varPi ^\textrm{fr}_{\ell _\text {c}}$ (green), and the portion of viscous energy that does not enter the fracture driving force $\int \limits _{ \varOmega _0} D^\textrm{vi} \; \textrm{d} V$ (red). For approach B ($\beta _\text {vi}=1$), the curves for $W^\textrm{ext}$ and $\varPi _{{\ell _\text {c}}}$ coincide. (Color figure online)

The length scale parameter ${\ell _\text {c}}$ is set to ${\ell _\text {c}}={0.275}{\hbox {mm}}$, which is identical to [42]. In order to enable a step-by-step analysis of the model, a constant fracture toughness is assumed, here, and ${\mathcal {G}}_\text {c}(\textbf{d})$ according to (39) is investigated in Sect. 4.3. Furthermore, with the aim of performing a thorough analysis of viscous fracture driving force contribution in Sect. 4.2, the two limiting cases $\beta _\text {vi}= 0$ (approach A) and $\beta _\text {vi}=1$ (approach B) are considered. Under these two assumptions, the respective values of ${\mathcal {G}}_\text {c}$ are identified from experimental data. For this purpose, regarding the critical deformation in SENT for the two rates experimentally investigated, deviation between simulation and mean values from the experiments is minimised by means of a gradient-free approach. Since the specimens are of low thickness, plane stress conditions are assumed and two-dimensional simulations are performed, here. Due to symmetry, only one half of the SENT specimen is considered. The mesh consists of quadratic triangular elements and is refined along the crack path. h-convergence is verified. The optimal simulation results are compared to the range of experimental data in Fig. 7 and the parameters of the fracture model are summarised in Table 3. Furthermore, the evolution of the relevant energetic quantities during the SENT is shown in Fig. 8, for which the lower displacement rate $\dot{{\bar{u}}} = {25}\,{\hbox {mm/min}}$ is considered exemplary. For both $\beta _\text {vi}=0$ with optimal ${\mathcal {G}}_\text {c}= {10.7}\,{\hbox {N/mm}}$, and $\beta _\text {vi}=1$ with optimal ${\mathcal {G}}_\text {c}= {12.0}\,{\hbox {N/mm}}$, good agreement between simulation and experiment can be stated. With $\beta _\text {vi}=1$, a marginally better approximation is obtained for this setup. However, in both cases, the critical force is slightly overestimated.¹⁶ Furthermore, especially for the higher rate $\dot{{\bar{u}}} = {200}\,{\hbox {mm/min}}$, the simulated F–u curves do not completely reproduce the smooth decrease experimentally observed in the post-critical stage preceding complete failure. Instead, the critical point is followed by a sudden drop of reaction force that, interestingly, does not come along with complete failure yet. It corresponds to crack initiation at the tip of the pre-existing notch, see Fig. 9, and is succeeded by a smoother decrease of force for which crack propagation through the specimen involves a slight increase of external load before, finally, it comes to complete failure.¹⁷ To the best of the author’s knowledge, such a phenomenon does not arise in hyperelastic models, whereas it also has been reported for linear viscoelasticity [42, 50]. The effect is the more pronounced the lower $\dot{{\bar{u}}}$. Apparently, it is provoked by the rate-dependent behaviour of the bulk material that involves an increase of effective stiffness as well as the effective load bearing capacity of the material when, locally in the vicinity of the crack, rate of deformation suddenly raises up due to the initiation of fracture. From an energetic point of view, as it can be seen from Fig. 8, the initiation of the phase-field crack at the critical point comes along with a sudden drop of free energy $\varPi ^\textrm{sd}$ stored within the specimen and the fracture pseudo-energy $\varPi ^\textrm{fr}$ raises, accordingly. In the post-critical range, when the crack propagates through the specimen, $\varPi ^\textrm{fr}$ continues to increase, whereas $\varPi ^\textrm{sd}$ does also raise directly after the critical force level, yet starts to decrease when the displacement load ${\bar{u}}$ approaches its maximum value. This behaviour is immaterial of the assumption made for $\beta _\text {vi}$, i.e. for the portion of viscous dissipation entering the fracture driving force. For a rigorous analysis within the small strain context, the reader is referred to the previous work [50].

Fig. 9

SENT—crack propagation through the specimen for $\dot{{\bar{u}}}={200}\,{\hbox {mm/min}}$ and $\beta _\text {vi}=0$ (approach A). The corresponding force–displacement curve is depicted in Fig. 7. Qualitatively similar results are obtained for approach B and other $\dot{{\bar{u}}}$

4.2 Model validation and analysis of viscous driving force

For further model validation and analysis, double notched specimens under tension (DENT) with varying length of the pre-existing notch z are considered as depicted in Fig. 10.

Fig. 10

DENT—setup for model validation and analysis. For comparison of model prediction with experimental data from [42], symmetrical specimens are considered, i.e. $m={75/2}\,{\hbox {mm}}$

At first, a symmetrical specimen geometry is considered, i.e. $m={75/2}\,{\hbox {mm}}$. The predictions of the model parameterized in the previous Sect. are compared to experimental data from [42] for $z \in \{9, 5\} \, \textrm{mm}$ and a constant rate $\dot{{\bar{u}}}={75}\,{\hbox {mm/min}}$ in Fig. 11. For both approaches A and B, model predictions fit the experimental results well, which is also true for $z \in \{7, 3\} \, \textrm{mm}$ (not depicted). The good agreement demonstrates the predictive capability of the present model and the suitability of the parameter identification from experiments with homogeneous and single-notched specimens.

Fig. 11

Symmetrical DENT—experimental data [42] versus model prediction for two values of length of pre-existing notch $z \in \{9,5\} \, \textrm{mm}$ and constant $\dot{{\bar{u}}}$. Similar results are obtained for $z= {7}\,{\hbox {mm}}$, see Fig. 12, and $z={3}\,{\hbox {mm}}$ (not depicted)

With the aim of more thoroughly analysing the rate-dependency of responses and elaborating on the driving force contributions, additional simulations are performed for $z= {7}\,{\hbox {mm}}$ and various rates $\dot{{\bar{u}}} \in [12.5, 400] \, \,{\hbox {mm/min}}$. The numerical predictions for the two approaches $\beta _\text {vi}=0$ and $\beta _\text {vi}=1$ are compared in Fig. 12.

Fig. 12

Symmetrical DENT—comparison of model prediction for approaches A and B for various rates and a fixed size of pre-existing notch $z= {7}\,{\hbox {mm}}$

Regardless of the approach for the driving force, for high displacement rates, the responses converge against an upper elastic limit for which there is almost no viscous dissipation until failure. For very low $\dot{{\bar{u}}}$, the responses of the structure likewise approach a lower elastic limiting case where over-stresses do approximately vanish during entire simulation. In between, for intermediate displacement rates, the critical displacement level diminishes with $\dot{{\bar{u}}}$ for both approaches A and B. In contrast, regarding the rate-dependency of critical force level, the model predictions do significantly differ depending on whether a viscous fracture driving force contribution is assumed or not. On the one hand, critical force monotonically increases with rate when there is no such contribution, i.e. $\beta _\text {vi}=0$ (A). On the other hand, for $\beta _\text {vi}=1$ (B), critical force becomes minimal for intermediate $\dot{{\bar{u}}}$, for which the greatest critical values of $\varPsi ^\text {vi}$ are observed, see Fig. 13. Although no experimentally-determined force–displacement curves are available, it can be stated that the former is in agreement with experimental observations [42], whereas the latter contradicts experimental experience. At least when modelling fracture of elastomeric materials under monotonic loading, in some cases, fracture driving force contribution associated to accumulated viscous dissipation can thus lead to erroneous model predictions. In other words, modelling approach A has revealed more plausible, which, in a sense, is different from plasticity, where a fracture driving force related to inelastic mechanisms has revealed advantageous [30, 39]. Interestingly, such an observation has not been made in the previous study within the small strain framework [50], where a less pronounced influence of viscous effects on crack propagation has been observed. This can probably be attributed to the fact that the present formulation enables to describe larger deviations away from thermodynamic equilibrium, resulting in considerably greater viscous contributions to fracture driving force.

As it has been comprehensively described in [50], it essentially is the change of effective stiffness and the amount of dissipation until failure that lead to the change of critical force and displacement level with rate of external load. While the amount of fracture driving force necessary for crack growth remains constant, the fracture driving force available for a constant level of deformation can change with rate. On the one hand, effective stiffness of the viscoelastic material monotonically increases with increasing rate of deformation. For a certain external displacement ${\bar{u}}$ prescribed, the density of strain energy raises with $\dot{{\bar{u}}}$, accordingly. On the other hand, in case of monotonic loads, the amount of viscous dissipation and thus, in case of $\beta _\text {vi}>0$, the level of $\varPsi ^\text {vi}$ at failure becomes maximal for intermediate rates.

Fig. 13

Symmetrical DENT—free energy contribution related to viscous dissipation (approach B) for various rates and a fixed size of pre-existing notch $z= {7}\,{\hbox {mm}}$

Although viscous fracture driving force contribution has revealed not advantageous for describing failure of elastomers under monotonic loads, it might be suitable for other classes of materials, e.g. thermoplastics, and especially for the modelling of fatigue fracture, e.g. with $0 < \beta _\text {vi}\ll 1$. In composites and thermoplastic materials, for instance, viscous dissipation and self-heating mechanisms can have an important influence on fatigue life, cf. [80].¹⁸ Furthermore, the viscous dissipative fracture driving force contribution may also be more suitable for materials that show some fluid-like properties, so that similar to plasticity, inelastic permanent deformations play a more important role.

Crack patterns in asymmetrical specimens In addition to the symmetrical specimens, simulation results are presented in the following for an asymmetrical DENT geometry as depicted in Fig. 10 with $m={27.5}\,{\hbox {mm}}$ and $z={9}\,{\hbox {mm}}$. Since for ductile fracture of metals, where instead of viscoelasticity another class of dissipative materials is involved, the choice of fracture driving force revealed crucial the appropriate numerical description of asymmetrical crack patterns, cf. [30], simulations are performed for both approaches A and B. The corresponding force–displacement curves are depicted in Fig. 14. The overall rate-dependency of the specimen response is identical to what has been described above for the symmetrical geometry. In particular, for $\beta _\text {vi}=1$, the numerically predicted critical force becomes minimal for an intermediate rate of external displacement, which does hardly coincide with what would be observed in experiments. In Fig. 15, the final crack patterns are compared for $\dot{{\bar{u}}}={200}\,{\hbox {mm/min}}$. In order to ease comparison, the phase-field is shown with respect to the reference domain $\varOmega _0$. For both fracture driving forces A and B, the crack pattern predicted for the viscoelastic material is essential different from what is typically observed when metals fail in a ductile manner. Instead of a single crack that connects the two pre-existing notches, two cracks independently propagate through the specimen. At a certain length, one of the two stops to propagate, resulting in an asymmetrical final crack pattern, see Fig. 16. Regardless of $\beta _\text {vi}$ and $\dot{{\bar{u}}}$, qualitatively identical crack paths are predicted.¹⁹ However, depending on $\beta _\text {vi}$, slight differences concerning the final length of the shorter crack can be stated especially for intermediate rates. Interestingly, when critical force is reached, the two cracks suddenly propagate over a finite width, which comes along with a significant abrupt drop of force. For intermediate and higher rates, similar to SENT geometry, a slight increase of external displacement ${\bar{u}}$ is necessary to make one of the cracks propagate further, resulting in a less heavy slope of the force–displacement curve before it finally comes to catastrophic failure. For these higher rates, in the simulations there is a stage that can be seen as a kind of stick–slip-like crack propagation, where the crack tip suddenly advances over a finite distance and then arrests over and over again. Alternatively, this stick–slip behaviour can also be seen as a series of subsequent initiations of small cracks.²⁰

These effects also lead to a non-smooth F–u curve in the post-critical range. Interestingly, for very small $\dot{{\bar{u}}}$, such a behaviour is not simulated. In the literature on dynamic crack growth, comparable phenomena have been reported, cf. e.g. [83]. However, it has to be noted that regarding this particular aspect, the predictive capabilities of the present model are somewhat limited, as inertia effects are not taken into account.

Fig. 14

Asymmetrical DENT—comparison of specimen responses for approaches A and B for various rates, pre-notch position $m={27.5}\,{\hbox {mm}}$ and notch length $z= {7}\,{\hbox {mm}}$

Fig. 15

Asymmetrical DENT—final crack patterns in the reference configuration $\varOmega _0$ for approaches A and B and $\dot{\bar{u}} = {200}\,{\hbox {mm/min}}$. For the EPDM rubber considered, no experimental results are available for this setup. Nevertheless, the crack paths resemble experimental observations made for other viscoelastic materials such as the hydroxyl-terminated polybutadiene (HTPB)-based solid propellant investigated in [82]

Fig. 16

Asymmetrical DENT—phase-field crack initiation and propagation through the specimen for $\dot{{\bar{u}}}={200}\,{\hbox {mm/min}}$ and $\beta _\text {vi}=0$ (approach A). The corresponding force–displacement curve is depicted in Fig. 14. Qualitatively similar results are obtained for approach B and other $\dot{{\bar{u}}}$

For the EPDM rubber for which the model has been parameterized here, no experimental results are available for crack propagation in asymmetrical specimens. Nevertheless, the crack patterns simulated with the present model are in excellent agreement with what has been observed in experiments for other viscoelastic materials, see e.g. [82]. It is obvious that, when specimen geometries are similar, these crack patterns in viscoelastic materials can differ from the ones that form in elasto-plastic ones, since the inelastic mechanisms are essentially different. For example, there typically is no zone of inelastic localisation in viscoelastic materials whereas localisation of plastic deformation can play an important role when it comes to ductile fracture of metals.

Creep fracture In addition to fracture under monotonically increasing loads, a qualitative analysis of creep fracture is performed by means of one representative example. For this purpose, the symmetrical DENT geometry with $z={7}\,{\hbox {mm}}$ is revisited. Instead of ${\bar{u}}$, a traction force ${\overline{F}}$ is applied that linearly increases with time until a certain value ${\overline{F}}_\textrm{max}$ is reached and is hold constant, subsequently. For two different values of ${\overline{F}}_\textrm{max}$, boundary conditions and model predictions are depicted in Fig. 17 for both approaches A and B. It can be stated that, generally, creep fracture can be captured regardless of the value of the assumption made on fracture driving force.²¹ In case of $\beta _\text {vi}>0$, failure can occur for lower ${\overline{F}}_\textrm{max}$ and after a shorter amount of creep time than for $\beta _\text {vi}=0$. Furthermore, if a fracture driving force contribution from viscous dissipation is assumed, it can also depend on the rate $\dot{{\overline{F}}}$ if creep fracture is predicted, since viscous dissipation vanishes for very small $\dot{{\overline{F}}}$, see [50] for a discussion in the small strain context.

4.3 Investigation of rate-dependent fracture toughness

In the foregoing section and the previous work [50], it is demonstrated that within the scope of an energetic phase-field fracture approach, a rate-dependent material model for the bulk induces a certain relationship between critical load and rate of deformation when ${\mathcal {G}}_\text {c}$ is constant. Therefore, in addition to experimental indication [2, 3, 52, 53, 73], there also is a clear motivation for assuming a rate-dependent toughness from a phenomenological point of view. Assuming ${\mathcal {G}}_\text {c}$ to be a function of effective rate of deformation $r = \Vert \textbf{d}\Vert _\textrm{F}$ enables more flexibility in describing the rate-dependent failure of varied materials. In what follows, this is demonstrated by means of numerical studies considering both an increase and a decrease of ${\mathcal {G}}_\text {c}$ with r. For this purpose, the symmetrical DENT setup with $z={7}\,{\hbox {mm}}$ and $\beta _\text {vi}=0$ is revisited. For ${\mathcal {G}}_\text {c}$, the sigmoid-shaped function (39) is assumed with ${\mathcal {G}}_\text {c}^1={10.7}\,{\hbox {N/mm}}$ and $\beta _\text {vi}=0$ as parameterized for EPDM whereas the responses for different ${\mathcal {G}}_\text {c}^2>{\mathcal {G}}_\text {c}^1$ as well as ${\mathcal {G}}_\text {c}^2< {\mathcal {G}}_\text {c}^1$ are investigated. Apart from that, the parameters are identical to the ones listed previously.

Fig. 17

DENT—boundary conditions and model predictions for the investigation of creep fracture

The case of ${\mathcal {G}}_\text {c}$ increasing with rate of deformation is investigated first. As a representative example, the specimen response is depicted in Fig. 18 for ${\mathcal {G}}_\text {c}^2= 2 \, {\mathcal {G}}_\text {c}^1$, $r_\text {ref}={2}\,{s^{-1}}$, $c=10/r_\text {ref}$. For this specific choice of $r_\text {ref}$, before it comes to crack propagation, the effective rates of deformation r satisfy $r \ll r_\text {ref}$ within the entire domain for all $\dot{{\bar{u}}} \lesssim {300}\,{\hbox {mm/min}}$. Through comparison of Figs. 18 and 12, it becomes clear that for these smaller rates, the pre-critical range of the specimen response is identical to the case where ${\mathcal {G}}_\text {c}= {\mathcal {G}}_\text {c}^1= \text {const}.$ In particular, effective stiffness and critical force raise with rate $\dot{{\bar{u}}}$, whereas critical deformation decreases. When the critical point is reached and crack propagation starts, effective rate of deformation r suddenly raises up within the material, resulting in an increase of ${\mathcal {G}}_\text {c}\left( r(\textbf{d}) \right)$. Accordingly, in the post-critical range of the F–u curves, a slightly less sharp slope can be observed with respect to ${\mathcal {G}}_\text {c}= \text {const}.$ However, this effect is not very pronounced compared to the effects arising from the rate-dependent toughness when pre-critical rate of deformation r becomes close to the threshold value $r_\text {ref}$.²² In that case, deformation at failure begins to raise with rate similar to stiffness and critical force. Experimentally, similar effects can be observed for some natural materials, see e.g. [84] for an overview, as well as viscoelastic silicone elastomer based model systems [85].

Fig. 18

DENT—model prediction in case of fracture toughness ${\mathcal {G}}_\text {c}$ assumed to increase with effective rate of deformation r

Fig. 19

DENT—model prediction in case of fracture toughness ${\mathcal {G}}_\text {c}$ assumed to decrease with effective rate of deformation r

For the discussion of ${\mathcal {G}}_\text {c}$ decreasing with r, ${\mathcal {G}}_\text {c}^2={\mathcal {G}}_\text {c}^1/4$, $r_\text {ref}={2}\,{s^{-1}}$, $c=5/r_\text {ref}$, are considered, exemplary. From the force–displacement curve depicted in Fig. 19 it appears that for $\dot{{\bar{u}}} \ll {300}\,{\hbox {mm/min}}$, the responses do again coincide with the case ${\mathcal {G}}_\text {c}= {\mathcal {G}}_\text {c}^1= \text {const}.$ Naturally, the initiation of the phase-field crack at the notch tips is immediately followed by complete failure, since in this moment, the sudden increase in rate of deformation comes along with a drop of toughness. Nevertheless, for the DENT geometry, similar behaviour is obtained as simulation result for ${\mathcal {G}}_\text {c}=\text {const}.$, which is in agreement with experiments. For high rates $\dot{{\bar{u}}} \ge {400}\,{\hbox {mm/min}}$, where $r \gtrsim r_\text {ref}$ also holds in pre-critical range, the decrease of deformation of failure that stems from the rate-dependent stiffness of the viscoelastic material is further intensified by the rate-dependent fracture toughness. In addition, the critical force does no longer raise up with $\dot{{\bar{u}}}$ yet also decreases. For sugar-based confections [86], a similar characteristic behaviour has been observed very recently. For high displacement rates, these materials fail in a brittle manner, i.e. at small deformation as well as low external force, whereas at low rates, they can undergo large deformation.²³

5 Conclusion and outlook

For the simulation of fracture of materials with rate-dependent behaviour, a flexible phase-field model is presented. To this end, the theory of finite viscoelasticity [46] is adopted for the deformation of the bulk material. The phase-field model is formulated such that, depending on the choice for the parameters, a portion of viscous dissipation can enter the fracture driving force. Moreover, in addition to the viscoelastic model of the bulk material, a fracture toughness function that depends on rate of deformation can be considered.

In order to analyse the coupling between different rate effects, a gradual analysis of the model is performed. The model of finite viscoelasticity is parameterized for an EPDM rubber based upon stress-deformation curves from the literature. Ogden-type strain energy densities are considered for both the equilibrium and over-stress parts of the response and very good agreement of the model with experimental data is obtained. Assuming a constant fracture toughness for the EPDM rubber, two limiting cases are studied regarding the fracture driving force and the respective values of toughness are identified from experimentally-determined SENT force–displacement curves. In doing so, either entire viscous dissipation or only effectively stored strain energy is assumed to enter the fracture driving force, respectively. In the absence of a driving force contribution related to viscous dissipative mechanisms, very good agreement between model predictions and experiments can be stated for different setups. In this case, plausible results are obtained over a broad range of rates of external load and deformation, respectively. On the contrary, if viscous dissipation is assumed to enter fracture driving force, erroneous model predictions can arise, here. In this case, agreement with experimental data is obtained for some specific rates, only. Accordingly, different from e.g. phase-field modelling of ductile fracture in metals, a distinct fracture driving force contribution related to inelastic dissipative mechanisms as proposed in [42, 43] or [40] has revealed not favourable for viscoelastic materials, in particular not for rubbery polymers. Furthermore, comparing the crack paths predicted in asymmetrical DENT specimens, it is demonstrated that such a driving force contribution is not necessary in order to predict non-symmetric crack patterns in an appropriate manner.

By means of a numerical study, it is demonstrated that a rate-dependent fracture toughness can significantly increase the capability of the phase-field model in capturing varied experimentally-observable responses. In particular, it seems suitable to describe rate-dependent brittle-to-ductile fracture mode transitions. In contrast, in case of a constant toughness, the rate-dependent model of bulk deformation induces a certain rate-dependency of critical stress and deformation, which does not coincide with experimental evidence for some specific materials. At least from a phenomenological point of view, rate-dependent fracture toughness thus seems to be an essential tool for modelling of rate-dependent fracture phenomena. While this contribution clearly demonstrates the potential of a rate-dependent fracture toughness within the proposed model, a quantitative description of rate-dependent brittle-to-ductile fracture mode transitions is beyond its scope. A thorough experimental analysis of these effects in materials with rate-dependent deformation behaviour, e.g. caramel-based confections [86], as well as a quantitative description based upon the framework presented in this contribution are the subject of current work.

Appendices

Appendix

1.1 A Tangent for the local Newton iteration

For the iterative solution of the viscous evolution equation (61) in the corrector step, the derivatives

(76)

with²⁴

(77)

(78)

are required. For the plane stress case, in addition, the derivatives

$$\begin{aligned} \frac{\partial \, r_\varrho }{\partial \varepsilon _3} = - \delta _{\varrho 3} \text {,}\end{aligned}$$

(79)

(80)

(81)

have to be evaluated.

B Material tangent

The consistent Lagrangian material tangent

(82)

is determined in order to enable the iterative solution of the weak form of balance of mechanical equilibrium (72). In line with e.g. [87], the derivation of the tangent is performed assuming $N_\lambda ={N^\textrm{el}_\lambda }=3$ and the case of identical principal stretches or elastic principal stretches is then a posteriori addressed by means of L’Hôpital’s rule. The equilibrium part of the virtually undamaged tangent is given by

(83)

wherein $\varvec{N}_\varrho$ denote the orthonormal eigenvectors of $\textbf{C}$,²⁵ see e.g. [68, 69] or [88] for a derivation of the derivatives of principle stretches and projection tensors. Into this expression (83), for the specific model under consideration,

(84)

can be inserted. In case of multiple principal stretches, i.e. $\exists \beta \ne \alpha : \lambda _\beta = \lambda _\alpha$, the second term in (83) can be evaluated making use of L’Hôpital’s rule [87]

(85)

Following [46], for the derivation of , a virtually undamaged over-stress tensor

(86)

is introduced with reference to the intermediate configuration defined by the viscous deformation gradient of the previous time step , i.e. based on the decomposition of the deformation gradient at increment n into

(87)

With

$$\begin{aligned} {\breve{\textbf{C}}}^\textrm{el}_\textrm{tr}= {{\textbf{F}}^\textrm{el}_\textrm{tr}}^\top \cdot {{\textbf{F}}^\textrm{el}_\textrm{tr}}\text {,}\end{aligned}$$

(88)

the over-stress part of the virtually undamaged material tangent then can be written as

(89)

wherein the Einstein summation convention applies for double indices. From this, the over-stress tangent in terms of the intermediate configuration described by can be defined to

(90)

In analogy to (83), this contribution to the material tangent is given by

(91)

wherein $\breve{\varvec{N}}_{\varrho }$ denote the orthonormal eigenvectors of ${\breve{\textbf{C}}}^\textrm{el}_\textrm{tr}$ and are the eigenvalues of . The first term in (91) can be rewritten making use of

(92)

Furthermore, for the derivatives with respect to the trial stretch quantities, use of

$$\begin{aligned} 0 = \frac{\partial r_\varrho }{\partial {\varepsilon _{\textrm{tr} \, {\sigma }}^\textrm{el}}} \text {,}\end{aligned}$$

(93)

which holds if the local Newton iteration has converged towards zero, is made. This assumption leads to

$$\begin{aligned} \frac{\partial {\varepsilon _{\varrho }^\textrm{el}}}{\partial {\varepsilon _{\textrm{tr} \, {\sigma }}^\textrm{el}}} = \left( \frac{\partial r_\sigma }{\partial {\varepsilon _{\sigma }^\textrm{el}}} \right) ^{-1} \text {,}\end{aligned}$$

(94)

which is given by (76) and further specified in Appendix A. Accordingly, the derivative in (92) can be expressed as

(95)

with

(96)

and the two contributions in (96) specified by (77) and (78). In case of multiple elastic principal stretches, i.e. $\exists \beta \ne \alpha : {\lambda _{\beta }^\textrm{el}} = {\lambda _{\alpha }^\textrm{el}}$, for the treatment of the second term in (91), the same procedure applies as outlined above for the case of multiple principal stretches. In particular, L’Hôpital’s rule reads

(97)

with the respective derivatives given by (92).