Damage Behaviour of Quasi-Brittle Composites: Mathematical and Computational Aspects

Abstract

In the present paper, an evaluation of the damage behaviour of quasi-brittle composites exposed to mechanical, thermal, and other loads is studied by means of viscoelastic and/or viscoplastic material models, applying some non-local regularisation techniques to the initiation and development of damages. The methods above are presented as a strong tool for a deeper understanding of material structures in miscellaneous engineering disciplines like civil, mechanical, and many others. Nevertheless, all of the software packages reflect certain compromises between the need for effective computational tools, with parameters obtained from inexpensive experiments, within the possibilities and the complexity of both physical and geometrical descriptions of structure deformation within processes. The article is devoted to the mathematical aspects regarding a considerably wide class of computational modelling problems, emphasising the following ones: (i) the existence and the uniqueness of solutions of engineering problems formulated in terms of the deterministic initial and boundary value problems of partial differential equations theory; (ii) the problems of convergence of computational algorithms applied to (i). Both aspects have numerous references to possible generalisations and investigations connected with open problems.

1. Introduction

Numerous engineering materials can be characterised as composites, i.e., materials produced from two or more constituents. The classification of [1] distinguishes among (i) miscellaneous classes of ductile materials (see [2,3]); (ii) brittle materials, cf. [4], with potential transition to quasi-brittle behaviour in some cases, cf. [5]; and (iii) quasi-brittle materials, discussed in [6,7,8]. Apart from various mechanical characteristics of their constituents and their interfaces, typical particularly for matrices with stiffening fibres, there may appear some other phenomena commonly observable, particularly for materials from (iii). They are the initial, micro-fractured zones with growth potential that result in visible macro-cracks. Let us recall the general review of composite structures [9], considered as various kinds of anisotropic and heterogeneous structures, including composite-like laminates and fibre-reinforced matrices. Connecting this with the formal and variational asymptotic methods can be useful for further investigations, although the damage and failure analysis is generally viewed as a method or direction to be developed in the future.

The computations at electronic, atomistic, microscopic, mesoscopic, etc., scales, together with various approaches to scale bridging, are analysed in [10]. In recent studies, scale bridging appears as an inspiring method. It is applied on various levels, from working with multiscale discrete damages (see [11]) up to the molecular dynamics applied for sufficiently simple material structures in [12]. An advanced non-local deterministic multiscale damage model is presented in [13]. A survey and a mutual comparison of various computational approaches to fractures and damages of quasi-brittle materials, like the discrete crack and the smeared crack techniques, molecular dynamics, etc., considered in context with their history and perspectives, can be found in [14].

Nevertheless, the detailed data are rarely known in sufficient detail. This is the case of cementitious composites with steel reinforcement, significant in design and construction in civil engineering. Thus, qualitative and/or quantitative information on smaller scales are exploited as the motivation for the formulation of semi-empiric constitutive relations, e.g., those between strain and stress fields. On the other hand, the primary formulation of computational models relies on the conservation principles of classical thermodynamics, compatible with [15,16]. As a representative example of such a scale-bridging approach, the model of [17] introduces concrete as a two-phase composite material consisting of (i) coarse aggregates and (ii) mortar matrix; microcracks are allowed in (ii). The stress and strain fields are evaluated at the mesoscale, exploiting the classical Eshelby solution (see [18]) for equivalent inclusions. The crack density parameter is then interpreted as a scalar damage variable at the macroscopic scale. As an alternative to such an approach, predominantly probabilistic computational techniques of different types are considered in [19,20,21].

Due to the usefulness of such methods for civil engineering designs and constructions, the prediction of the damage behaviour of cementitious composites, including concretes with various kinds of reinforcement, is discussed in numerous studies. As for the progress in this field made in the last three decades, we mention the sequence of upgrades of the so-called Mazars model documented by [22,23,24,25,26,27]. The original Mazars model was designed for the description of the elastic damage behaviour of concrete. The model is three-dimensional in general and is tailored for isotropic materials. It works with a damage criterion expressed in terms of strain, taking the traction–compression asymmetry into account. Its later improvements cover the restoration of stiffness forced by the re-closing of cracks, viscous and/or plastic deformation effects and influences, etc. As for its comparison with investigations of details of crack branching, cf. [28], and as for an advanced model with a small length of nonlocality based on the formulation of Griffith fracture energy, cf. [29], such models work with some irreversible damage factor, usually nonlocal, with a sufficiently simple reversible strain–stress relation, e.g., a viscoelastic one. Some recent studies reflect new advances in adjacent research areas; see [30].

In this paper, we present work with assessments of the level of damage based on multiple factors; the methods are supported by particle swarm optimisation. There is also the paper of [31], where the effect of cyclic loading conditions is evaluated. There is another paper, that of [32], where the phase field as a modelling object is involved in consideration; this approach is motivated by [33]. Such new results inspire the development of a new generation of Mazars models, which is obvious from [34,35]. Another new approach comes from [36] and provides the rate-independent plastic strain from a sequence of quasi-static brittle damage evolutions; the model is based on the mathematical $\Gamma$-convergence theory.

The authors of this article and their collaborators at Brno University of Technology have quite a long experience in the development of viscoelastic, viscoplastic and other material models for engineering applications, including damage, namely (i) in the extended finite element and analysis of traction–separation laws on cohesive interfaces (V. Kozák), (ii) in generalised Kelvin chains, including inverse problems (M. Trcala), (iii) in the analysis of collisions of deformable bodies (I. Němec), (iv) in the applications of differential geometry to modern physics (J. Tomáš) and (v) in the mathematical verification of solvability of engineering problems and convergence of computational algorithms (J. Vala). Their recent results [37,38,39,40,41,42,43,44,45,46,47,48] are exploited in this article.

The present introductory Section 1 will be followed by Section 2, containing more comments on the physical background and engineering simplifications. The principal aim of this article consists in the mathematical verification of the solvability of a model problem, working with a viscoelastic body, sketched by Figure 1, without and with incorporated damage. This model problem will be introduced in Section 3 in a rather simplified way to avoid technical difficulties in proofs and deep studies of advanced (and in many cases incomplete) mathematical theories.

Table 1. The rough scheme of the proofs in Section 3.

Step	Topics
1	weak formulation of a model problem in appropriate Bochner–Sobolev spaces
2	time-discretised formulation of 1 with a finite number of m steps
3	step-by-step verification of solvability of 2
4	a priori bounds for sequences of approximate solutions, independent of m
5	weak and strong convergence properties of sequences by 4,
6	identification of limits by 5 with solution of 1

presents the rough overview of steps needed for proper verifications of formal mathematical existence and convergence results, which are open to generalisations in several ways. They are indicated in Section 4, including some unclosed problems with highlighted directions related to numerical modelling and simulations of advanced materials, structures and technologies, significant in civil, mechanical and other branches of engineering. A model mathematical problem presented in Section 3 will be introduced in a rather simplified way to avoid (i) technical difficulties in proofs and (ii) deep studies of advanced (and in many cases incomplete) mathematical theories. We continue with a sketch of proper verifications of formal mathematical existence and convergence results, open to generalisations in several ways. Section 5 is devoted to the overview of computational approaches, supplied with an illustrative numerical example. The final Section 6 resumes all presented investigations completed by short comments on perspectives of future research. The rather long list of references prefers essential research results achieved in several recent years, not yet included in review books and articles. To improve the reader’s orientation in this list,

Table A1. Overview on references.

Section	Item(s)	Topics
1	[1,2,3,4,5,6,7,8]	damage classification: ductile vs. brittle vs. quasi-brittle damage
1	[9]	review of models for composite structures
1	[10,11,12,13,14]	multiscale approaches: from electronics up to macroscopic scale
1	[15,16]	monographs of classical thermomechanics
1	[17,18]	example of scale bridging for concrete
1	[19,20,21]	probabilistic computational techniques
1	[22,23,24,25,26,27]	Mazars model for concrete
1	[28,29,30,31,32,33,34,35,36]	alternatives and reformulations of Mazars model
1	[37,38,39,40,41,42,43,44,45,46,47,48]	useful recent results of authors and their collaborators at BUT
2	[49]	approaches to damage modelling
2	[50,51,52]	models containing generalized viscoelastic Kelvin chains
3	[53,54,55,56]	functional analysis: traditional results
3	[57,58]	functional analysis: modern results
3	[59]	line geometry
3	[60]	method od lines for partial differential equations
3	[61,62]	functional analysis: auxiliary lemmas
3	[63,64,65,66,67,68,69,70]	nonlocal stresses and damage factor(s)
3	[71]	functional analysis: an auxiliary result
4	[72,73]	fundamentals of finite element method(s)
4	[74,75,76,77]	generalized/extended finite element methods
4	[78,79]	functional analysis: non-Lipchitzian domains
4	[80,81]	coupled failure engineering models
4	[82,83,84]	thermodynamic formulations based on Gibbs energy
4	[85]	theory of subdifferentials
4	[86]	thermodynamics of selected materials and processes
4	[87,88,89]	from small strains/stresses to finite geometry formulations
4	[90,91,92,93,94,95,96,97]	structured deformation: fundamentals of geometry and physics
4	[98,99,100,101]	structured deformation: auxiliary lemmas
4	[102,103,104,105,106]	from Boltzmann to Cosserat continuum
4	[107,108,109,110,111,112,113]	fundamentals of differential geometry
4	[114,115]	Weil bundles, Stiefel and Grassmann manifolds
5	[116,117,118]	numerical solution of nonlinear algebraic systems of equations
5	[119,120]	adaptive strategies with enrichment functions

is included in Appendix A.

2. Physical Background and Engineering Simplifications

As announced in Section 1, from the macroscopic point of view, it is natural to work with the conservation of scalar quantities such as mass, linear and angular momentum components, and/or energy. As for our model problem, we can work only with the conservation of the linear momentum formulated for the Boltzmann continuum under the usual small strain simplification as follows. All quantities are related to the initial state, for which no irreversible damage occurs. The later remarks in Section 4 will be devoted to potential generalisations of such an approach.

The principal challenge of computational modelling of fracture in solid materials and structures is in the incorporation of complicated crack topologies. The fracture evolution can be attributed to the interplay between the crack topology and the distribution of deformation energy. Two main groups of numerical approaches, reviewed by [49], are as follows: (i) topology-modifying approaches and (ii) field-modifying approaches. In (i), it is necessary to make some steps identical or similar to remeshing, while in (ii), additional field(s) representing the spatial location of the fracture are required. Unfortunately, both (i) and (ii) are computationally expensive, namely for three-dimensional simulations with crack branching and interactions. In addition to that, their reliability strongly depends on the design and identification of material parameters from appropriate experiments.

Roughly speaking, we can recognise four stages of damage development from the engineering point of view. They are as follows: (a) a formation of microfractured zones, (b) an initiation of macroscopic cracks and their gradual growth, (c) a formation of systems of macroscopic cracks and their coalescence, (d) the total destruction of a considered material specimen or structure. From the point of view of solid-state physics, it is challenging to study the development from (b) to (c) and (d) thoroughly. Nevertheless, engineering practice needs, most of all, to incorporate (a) and evaluate the risk of progressive crack growth in (b), without extremely long and expensive distributed, parallel, etc., computations. This justifies the following approach, starting with a viscoelastic model based on the Kelvin and Maxwell chains, as introduced by [50], with later implementation of stiffness reduction due to (a), as demonstrated in Section 3, and potential activity of (b), sketched in Section 4.2.

For the sake of illustration, Figure 1 shows quite a simple three-component viscoelastic scheme. It is the well-known SLS (standard linear solid) or the Zener scheme, viewed as the analogy of serial, parallel, etc., schemes in the theory of electrical circuits. The related symbols $\mathcal{C}$, $\alpha$, $\beta$, $\sigma$, and $\tau$ will be introduced in Section 3 properly. For the preliminary information, $\mathcal{C}$ refers here to a symmetric matrix of material parameters for the Hooke law of purely elastic material behaviour, with up to 21 independent elements in the general case. This can be reduced to a certain function of two parameters in the particular isotropic case. The parameters are the well-known Lamé constant or, alternatively, the Young’s modulus and the Poisson ratio, apart from potential material inhomogeneity. Moreover, we define $\alpha$ and $\beta$ as certain additional material parameters, taking into account the more complicated viscoelastic behaviour. For the sake of simplicity, they are considered as two scalars. Further, $\tau$ and $\sigma -\tau$ denote two different kinds of stress in two parallel branches of the scheme. Thus, $\sigma$ can be understood as the total stress.

Such linear models (without damage) can be naturally extended to, e.g., nonlinear frequency–amplitude models, applying essentially [51]. A larger class of linear viscoelastic models, containing Kelvin chains, with more components, is introduced in [45,46]. Nevertheless, for an increasing number of their components or for their infinite number in the limit passage, any reasonable experimental setting of their quantitative characteristics becomes a non-trivial auxiliary optimisation problem. Nevertheless, there are still other types of model components, which can be implemented into the nonlinear Prandtl–Ishlinskiǐ operator by [52].

Notice that the small strain simplification, as introduced above, can be limiting for dynamic damage scenarios, namely for the simulation of contacts/impacts of deformable bodies, as discussed in [40]. Moreover, such simulation, as well as the proper analysis of energy dissipation on such contacts/impacts, can be seen as an additional serious problem. However, with regard to the quasi-static formulation of both versions of a model problem in Section 3, this simplification seems to be acceptable; for its removal, cf. Section 4.4.

3. A Model Problem

Following the necessary overview of mathematical preliminaries divided into two steps, we present the model problem in its weak formulation in the Lebesgue, Sobolev and Bochner–Sobolev spaces. In the first step of the mathematical preliminaries, we investigate SLS free of any incorporated damage, while in the second step, we incorporate it. In particular, the elastic component in the upper branch of Figure 1 is missing for the classical Kelvin model. This situation corresponds to $\alpha =0$ fulfilled everywhere. It is easy to see that the second step requires some regularisation techniques, which is enforced by the loss of linearity.

3.1. Mathematical Preliminaries

In the first step, we conduct an analysis of the weak solvability of the linear parabolic (and later also hyperbolic, cf. Section 3.2) initial and boundary value problem. In the second step, we add a nonlocal and nonlinear term. The analysis of the elliptic linear problems can be found in [53,54], and it is adapted to parabolic and hyperbolic problems in [55]. A generalisation covering a large class of nonlinear problems can be found in [56]. In the present paper, we follow the notations and results from the monograph [57], completed by the regularisation results from [58].

Some ideas in proofs related to our model problem are similar to those from [41], where the classical viscoelastic Kelvin model is implemented. Nevertheless, we shall see that the generalisation of these results, needed in engineering applications like [46], is not straightforward and requires a separate mathematical analysis.

A deformable body is assumed to occupy a certain domain $\Omega$ with Lipschitzian boundary $\partial \Omega$ in the three-dimensional Euclidean space ${\mathbb{R}}^3$, supplied by a Cartesian coordinate system $x=(x_1,x_2,x_3)$, with $\partial \Omega$ consisting of two disjoint parts $\Gamma$ (for Neumann boundary conditions, inhomogeneous in general) and $\Theta$ (for homogeneous Dirichlet boundary conditions). An additional assumption of the non-zero Hausdorff measure of $\Theta$ on $\partial \Omega$ enables us to avoid the existence of infinitely many solutions (for equilibrium loads caused by an insufficient support) as well as the non-existence of any solution (for non-equilibrium loads caused by missing dynamic considerations). A strain and stress development on $\Omega$ will be considered for any time t belonging to a time interval $\mathcal{I}=[0,\varsigma ]$. The finite real $\varsigma$ opens the door to potential studies of a limit passage $\varsigma \to \infty$, which is not discussed in detail. An initial condition can be prescribed for $t=0$.

Let us consider the following function spaces introduced in [57], Part 1.2: $\mathcal{H}=L^2{(\Omega )}^3$, $\mathcal{E}=L^2{(\Omega )}_{\mathrm{sym}}^{3\times 3}$, $\mathcal{G}=L^2{(\Gamma )}^3$, $\mathcal{Z}=L^{\infty }(\Omega )$, $\mathcal{M}=L^2{(\Omega )}_{\mathrm{sym}}^{(3\times 3)\times (3\times 3)}$ and

$$\mathcal{V}=\{v\in W^{1,2}{(\Omega )}^3:v=o\mathrm{on}\Theta \},$$

(1)

where $o=(0,0,0)$. In the last definition, the strict support in all directions excludes any rigid motion; for the relation to the line geometry, cf. [59].

For an unknown variable $u(x,t)$ belonging to $L^2(\mathcal{I},\mathcal{V})$ in the sense of [57], Part 7.1, which was introduced as a vector of displacements related to the initial configuration at $t=0$, we are allowed to assume $u(x,0)=o$ for any $x\in \Omega$. As is usual in the linear theory of elasticity, we can introduce a strain tensor $\epsilon \left(v\right)\in \mathcal{E}$ for any $v\in \mathcal{V}$ with components ${\epsilon }_{ij}\left(v\right)=(\partial v_i/\partial x_j+\partial v_j/\partial x_i)/2$, where $i,j\in \{1,2,3\}$. As for the material characteristics, which are considered to be time-independent, it is natural to suppose that $\alpha ,\beta \in \mathcal{Z}$ and $\mathcal{C}\in \mathcal{M}$ satisfy $\alpha \left(x\right)\ge c$, $\beta \left(x\right)\ge c$, and $a^{\mathrm{T}}\mathcal{C}\left(x\right)a\ge c{a}^{\mathrm{T}}a$ for any $x\in \Omega$, any matrix $a\in {\mathbb{R}}_{\mathrm{sym}}^{3\times 3}$, and some positive constant c. All displacements u introduced above as well as the strains $\epsilon \left(u\right)$ and stresses $\sigma ,\tau \in L^2(\mathcal{I},\mathcal{E})$ (see Figure 1) are caused by the volume loads f on $\Omega$ and surface loads on $\Gamma$. To avoid unpleasant technical difficulties in proofs and to obtain stronger convergence results, we assume the smoothness in the sense of $f\in W^{1,2}(\mathcal{I},\mathcal{H})$ and $g\in W^{1,2}(\mathcal{I},\mathcal{G})$.

To simplify most of the formulae, let us apply the following notation for standard scalar products in the special Hilbert spaces: $(.,.)$ in $\mathcal{H}$, $\langle .,.\rangle$ in $\mathcal{G}$ $\left(\right(.,.\left)\right)$ in $\mathcal{V}$. In addition to that, for a fixed $t\in \mathcal{I}$ and an arbitrary $\gamma \in \mathcal{E}$, $\left[\right((\gamma ,\tau )\left)\right]$ must be understood in the sense of the Bochner integral of a special abstract function $\left((\gamma ,\tau (.,\tilde{t}))\right)$ over $\tilde{t}\in [0,t]$.

3.2. The First Step: A Viscoelastic Model

Let us consider an arbitrary test function $v\in \mathcal{V}$, which is the well-known concept of the vector of virtual displacements used in engineering mechanics. Further, consider $\tau \in \mathcal{H}$, which can be viewed as a symmetric matrix of certain virtual stresses where all of the values of $t\in \mathcal{I}$ are allowed. Then, the weak formulation of the linear momentum conservation reads

$$\left(\right(\epsilon \left(v\right),\sigma \left)\right)=(v,f)+\langle v,g\rangle .$$

(2)

The upper branch of Figure 1, stemming from the Maxwell model, corresponds to the integral constitutive relation

$$\left((\gamma ,{\mathcal{C}}^{-1}\tau /\alpha )\right)+\left[\left((\gamma ,{\mathcal{C}}^{-1}\tau /\beta )\right)\right]=\left((\gamma ,\epsilon \left(u\right))\right).$$

(3)

To involve the lower branch of Figure 1, it is sufficient to directly evaluate

$$\sigma =\tau +\mathcal{C}\epsilon \left(u\right)$$

(4)

on $\Omega$. This will be generalised in the second step. Inserting (4) into (2), we obtain

$$\left(\right(\epsilon \left(v\right),\tau \left)\right)+\left(\right(\epsilon \left(v\right),\mathcal{C}\epsilon \left(u\right)\left)\right)=(v,f)+\langle v,g\rangle .$$

(5)

Then, (5) together with (3) represents a weak formulation of a system of two partial differential equations of parabolic type, the unknowns of which are u and $\tau$. Both of them are time-dependent, in contrast to v and $\gamma$. The Neumann boundary condition is incorporated in the last additive right-hand-side term of (2) and (5). The Dirichlet boundary condition is involved in the definition of $\mathcal{V}$ by (1); natural Cauchy initial conditions are

$$u=o,\tau =\mathcal{O}$$

(6)

on $\Omega$. At this point, $\mathcal{O}$ denotes the zero-valued matrix from ${\mathbb{R}}^{3\times 3}$.

Nevertheless, the unknowns u and $\tau$ of this kind are elements of an infinite-dimensional abstract function space, and their analytical or at least semi-analytical form (expressed e.g., in the form of sums of certain a priori known function sequences) is rarely available. Thus, we need some discretisation technique applicable for practical computations. We come out from the method of the discretisation in time, following [57], Part 8. The time interval $\mathcal{I}$ will be covered by a finite number m of equidistant subintervals ${\mathcal{I}}_s^m$, introduced as the sets of all t satisfying $(s-1)h<t\le sh$ for $s\in \{1,\dots ,m\}$. Here, $h=\varsigma /m$ (this dependence of h on m is not expressed explicitly for brevity of notations); the analysis limit passage $m\to \infty$ is necessary.

Let us remind the reader that for $t=0$ (not contained in ${\mathcal{I}}_s^m$), we have (6). Nevertheless, the resulting sequence of partial differential equations of elliptic type, obtained step-by-step for individual s, is time-independent only formally, since they are still defined in the infinite-dimensional function space. From this point of view, the convergence considerations are still not constructive for practical evaluations, since finite-dimensional function spaces are required. This will be discussed in more detail in Section 5.

On the other hand, it is useful to see that the convergence of algorithms can be guaranteed without any additional assumptions on the discretisation of $\Omega$, $\Theta$, and $\Gamma$. Let us remind the reader of the concurrent approach, which is not preferred here. Namely, it is the Fourier multiplicative decomposition of unknown functions and the primary discretisation of $\Omega$, $\Theta$, and $\Gamma$. We refer to the so-called method of lines, which leads to the sparse (but still finite) systems of ordinary differential equations. Such systems require effective generalised eigenvalue analysis or an additional time discretisation. For a detailed overview, see [60].

The time-discretised formulation of (5), motivated by the famous Euler backward formula (designed for first-order ordinary differential equations originally), is

$$\left((\epsilon \left(v\right),{\tau }_s^m)\right)+\left((\epsilon \left(v\right),\mathcal{C}\epsilon \left(u_s^m\right))\right)=(v,f_s^m)+\langle v,g_s^m\rangle .$$

(7)

Here, for any fixed s and m, $f_s^m\in \mathcal{H}$ and $g_s^m\in \mathcal{G}$ can be considered as sufficiently good approximations of f and g on ${\mathcal{I}}_s^m$, e.g., as Clément quasi-interpolations by [57], Part 8.2, based on the generalised mean values on such abstract functions on ${\mathcal{I}}_s^m$, $u_s^m\in \mathcal{V}$, and ${\tau }_s^m\in \mathcal{H}$ refers to approximations of unknown values of u and $\tau$ valid on ${\mathcal{I}}_s^m$. Applying the rectangular rule for numerical quadrature, from (3), we analogously deduce

$$\left((\gamma ,{\mathcal{C}}^{-1}{\tau }_s^m/\alpha )\right)+h\left((\gamma ,{\mathcal{C}}^{-1}({\tau }_1^m+\dots +{\tau }_s^m)/\beta )\right)=\left((\gamma ,\epsilon \left(u_s^m\right))\right).$$

(8)

Clearly, (8) can be rewritten with $s-1$ instead of s. The difference between the original (8) and this modification gives

$$\left((\gamma ,{\mathcal{C}}^{-1}({\tau }_s^m-{\tau }_{s-1}^m)/\alpha )\right)+h\left((\gamma ,{\mathcal{C}}^{-1}{\tau }_s^m/\beta )\right)=\left((\gamma ,\epsilon (u_s^m-u_{s-1}^m))\right).$$

(9)

It is quite easy to observe that all of the notations under discussion are compatible with (6), i.e., ${\tau }_0^m=\mathcal{O}\ne$ formally, as well as $u_0^m=o$.

Since the system of (7) and (9) is linear for any fixed s and m, its solvability can be easily verified, applying the Lax–Milgram theorem, as it is formulated in [57], Part 2.3. The remaining ingredients, needed for the formal verification of all assumptions of this theorem, are (i) the standard Cauchy–Schwarz inequality (see [57], Part 1.1), (ii) the simplest version of the Sobolev compact embedding theorem together with its trace variant (see [57], Part 1.2), including their evident consequences in the form of estimates in norms of $\mathcal{H}$, $\mathcal{G}$, and $\mathcal{V}$:

$$(w,\tilde{w})\le {\parallel w\parallel }_{\mathcal{H}}\parallel \tilde{w}{\parallel }_{\mathcal{H}}\le {\parallel w\parallel }_{\mathcal{H}}{\parallel \tilde{w}\parallel }_{\mathcal{V}}$$

(10)

for each $w,\tilde{w}\in \mathcal{H}$ and

$$\langle w,\tilde{w}\rangle \le {\parallel w\parallel }_{\mathcal{G}}\parallel \parallel \tilde{w}{\parallel }_{\mathcal{G}}\le {c\parallel w\parallel }_{\mathcal{G}}\parallel \parallel \tilde{w}{\parallel }_{\mathcal{V}}$$

(11)

for each $w,\tilde{w}\in \mathcal{G}$, where c is a certain (sufficiently large) positive constant. We remark that its value is not essential at this point. This implies that the same symbol can be used for other generic constants of this kind as well. Lastly, we require (iii) the Korn inequality by [61]

$$\left((\epsilon \left(w\right),\epsilon \left(w\right))\right)\ge c^{-1}{\parallel w\parallel }_{\mathcal{V}}^2$$

(12)

for an arbitrary $w\in \mathcal{V}$. Moreover, ${\left\{f^m\right\}}_{m=1}^{\infty }$ and ${\left\{g^m\right\}}_{m=1}^{\infty }$ are two a priori known simple abstract functions mapping $\mathcal{I}$ to $\mathcal{H}$ and $\mathcal{G}$, whose values are just $f_s^m$, $g_s^m$ in $\mathcal{H}$ and $\mathcal{G}$, and the convergence properties of the so-called Rothe sequences ${\left\{f^m\right\}}_{m=1}^{\infty }$, ${\left\{g^m\right\}}_{m=1}^{\infty }$ to f and g are clear; cf. the discussion below (7). Unlike this, ${\left\{u^m\right\}}_{m=1}^{\infty }$ and ${\left\{{\tau }^m\right\}}_{m=1}^{\infty }$ are two a priori unknown simple abstract functions mapping $\mathcal{I}$ to $\mathcal{H}$ and $\mathcal{G}$, whose values are just $u_s^m$, ${\tau }_s^m$ in $\mathcal{V}$ and $\mathcal{E}$, but the analysis of convergence of ${\left\{u^m\right\}}_{m=1}^{\infty }$, ${\left\{{\tau }^m\right\}}_{m=1}^{\infty }$ is not trivial.

In particular, select $v=u_s^m-u_{s-1}^m$ in (7) and $\gamma ={\tau }_s^m$ in (9). Then, the sum of (7) and (9) yields

$$\begin{array}{c}\hfill \left(({\tau }_s^m,{\mathcal{C}}^{-1}({\tau }_s^m-{\tau }_{s-1}^m)/\alpha )\right)+h\left(({\tau }_s^m,{\mathcal{C}}^{-1}{\tau }_s^m/\beta )\right)+\left((\epsilon (u_s^m-u_{s-1}^m),\mathcal{C}\epsilon \left(u_s^m\right))\right)\\ \hfill =(u_s^m-u_{s-1}^m,f_s^m)+\langle u_s^m-u_{s-1}^m,g_s^m\rangle .\end{array}$$

(13)

The formal modification of (13) leads to

$$\begin{array}{c}\hfill \left(({\tau }_s^m,{\mathcal{C}}^{-1}{\tau }_s^m/\alpha )\right)/2-\left(({\tau }_{s-1}^m,{\mathcal{C}}^{-1}{\tau }_{s-1}^m/\alpha )\right)/2\\ \hfill +\left(({\tau }_s^m-{\tau }_{s-1}^m,{\mathcal{C}}^{-1}({\tau }_s^m-{\tau }_{s-1}^m)/\alpha )\right)/2\\ \hfill +h\left(({\tau }_s^m,{\mathcal{C}}^{-1}{\tau }_s^m/\beta )\right)+\left((\epsilon \left(u_s^m\right),\mathcal{C}\epsilon \left(u_s^m\right))\right)/2\\ \hfill -\left((\epsilon \left(u_{s-1}^m\right),\mathcal{C}\epsilon \left(u_{s-1}^m\right))\right)/2+\left(\left(\epsilon (u_s^m-u_{s-1}^m)\mathcal{C}\epsilon (u_s^m-u_{s-1}^m)\right)\right)/2\\ \hfill =(u_{s-1}^m,f_{s-1}^m-f_s^m)+(u_s^m,f_s^m)-(u_{s-1}^m,f_{s-1}^m)\\ \hfill +\langle u_{s-1}^m,g_{s-1}^m-g_s^m\rangle +\langle u_s^m,g_s^m\rangle -\langle u_{s-1}^m,g_{s-1}^m\rangle .\end{array}$$

(14)

Now, we can sum up all equations (14) for $s\in \{1,\dots ,r\}$, multiplied by 2, where r is an arbitrary fixed element from $\{1,\dots ,m\}$. The result is as follows:

$$\begin{array}{c}\hfill \left(({\tau }_r^m,{\mathcal{C}}^{-1}{\tau }_r^m/\alpha )\right)+h\sum _{s=1}^r\left(({\tau }_s^m-{\tau }_{s-1}^m,{\mathcal{C}}^{-1}({\tau }_s^m-{\tau }_{s-1}^m)/\alpha )\right)\\ \hfill +2h\sum _{s=1}^r\left(({\tau }_s^m,{\mathcal{C}}^{-1}{\tau }_s^m/\beta )\right)+\left((\epsilon \left(u_r^m\right),\mathcal{C}\epsilon \left(u_r^m\right))\right)\\ \hfill +\sum _{s=1}^r\left((\epsilon (u_s^m-u_{s-1}^m),\mathcal{C}\epsilon (u_s^m-u_{s-1}^m))\right)\\ \hfill =2(u_s^m,f_r^m)-2h\sum _{s=1}^r(u_{s-1}^m,D{f}_s^m)+2\langle u_s^m,g_r^m\rangle -2h\sum _{s=1}^r\langle u_{s-1}^m,D{g}_s^m\rangle \end{array}$$

(15)

exploiting the notation of relative differences $D{f}_s^m=(f_s^m-f_{s-1}^m)/h$, $D{g}_s^m=(g_s^m-g_{s-1}^m)/h$, usual in numerical differentiation. Moreover, for any positive $\eta$ and $\tilde{\eta }$, we have

$$\begin{array}{c}\hfill 2(u_s^m,f_r^m)=\eta (u_s^m,u_s^m)+{\eta }^{-1}(f_r^m,f_r^m)-(\eta u_s^m-{\eta }^{-1}{f}_r^m,\eta u_s^m-{\eta }^{-1}{f}_r^m),\\ \hfill 2\langle u_s^m,g_r^m\rangle =\tilde{\eta }(u_s^m,u_s^m)+{\tilde{\eta }}^{-1}\langle g_r^m,g_r^m\rangle -\langle \tilde{\eta }{u}_s^m-{\tilde{\eta }}^{-1}{g}_r^m,\tilde{\eta }{u}_s^m-{\tilde{\eta }}^{-1}{g}_r^m\rangle \end{array}$$

(16)

and also

$$\begin{array}{c}\hfill -2h\sum _{s=1}^r(u_{s-1}^m,D{f}_s^m)=\eta \sum _{s=1}^r(u_{s-1}^m,u_{s-1}^m)+{\eta }^{-1}\sum _{s=1}^r(D{f}_s^m,D{f}_s^m)\\ \hfill -\sum _{s=1}^r(\eta u_{s-1}^m+{\eta }^{-1}{f}_s^m,\eta u_{s-1}^m+{\eta }^{-1}{f}_s^m),\end{array}$$

(17)

$$\begin{array}{c}\hfill -2h\sum _{s=1}^r\langle u_{s-1}^m,D{g}_s^m\rangle =\tilde{\eta }\sum _{s=1}^r\langle u_{s-1}^m,u_{s-1}^m\rangle +{\tilde{\eta }}^{-1}\sum _{s=1}^r\langle D{g}_s^m,D{g}_s^m\rangle \\ \hfill -\sum _{s=1}^r\langle \tilde{\eta }{u}_{s-1}^m+{\tilde{\eta }}^{-1}{g}_s^m,\tilde{\eta }{u}_{s-1}^m+{\tilde{\eta }}^{-1}{g}_s^m\rangle .\end{array}$$

Inserting (16) and (17) into (15), we obtain

$$\begin{array}{c}\hfill \left(({\tau }_r^m,{\mathcal{C}}^{-1}{\tau }_r^m/\alpha )\right)+h\sum _{s=1}^r\left(({\tau }_s^m-{\tau }_{s-1}^m,{\mathcal{C}}^{-1}({\tau }_s^m-{\tau }_{s-1}^m)/\alpha )\right)\\ \hfill +2h\sum _{s=1}^r\left(({\tau }_s^m,{\mathcal{C}}^{-1}{\tau }_s^m/\beta )\right)+\left((\epsilon \left(u_r^m\right),\mathcal{C}\epsilon \left(u_r^m\right))\right)\\ \hfill +\sum _{s=1}^r\left((\epsilon (u_s^m-u_{s-1}^m),\mathcal{C}\epsilon (u_s^m-u_{s-1}^m))\right)\\ \hfill +\sum _{s=1}^r(\eta u_{s-1}^m+{\eta }^{-1}{f}_s^m,\eta u_{s-1}^m+{\eta }^{-1}{f}_s^m)+\sum _{s=1}^r\langle \tilde{\eta }{u}_{s-1}^m+{\tilde{\eta }}^{-1}{g}_s^m,\tilde{\eta }{u}_{s-1}^m+{\tilde{\eta }}^{-1}{g}_s^m\rangle \\ \hfill =\eta h\sum _{s=1}^r(u_{s-1}^m,u_{s-1}^m)+\tilde{\eta }h\sum _{s=1}^r(u_{s-1}^m,u_{s-1}^m)\\ \hfill +{\eta }^{-1}h\sum _{s=1}^r(D{f}_s^m,D{f}_s^m)+{\tilde{\eta }}^{-1}h\sum _{s=1}^r\langle D{g}_s^m,D{g}_s^m\rangle .\end{array}$$

(18)

The second, third, fifth, sixth, and seventh left-hand-side additive terms in (18) can be estimated by zero from below. The third and fourth left-hand-side additive terms are bounded from above by some positive constants, thanks to the interpolation properties of ${\left\{f^m\right\}}_{m=1}^{\infty }$ and ${\left\{g^m\right\}}_{m=1}^{\infty }$ of f and g. Taking into account (12) for the fourth left-hand-side additive term (the lower estimate for the first one is simple) and (10) and (11) for the first and second right-hand-side additive terms, we again obtain, for a sufficiently large positive constant c, the following:

$$\parallel {\tau }_r^m{\parallel }_{\mathcal{H}}^2+{\parallel u_s^m\parallel }_{\mathcal{V}}^2\le c\left(1+\sum _{s=1}^r{\parallel u_{s-1}^m\parallel }_{\mathcal{V}}^2\right).$$

(19)

Thanks to (19), the discrete Gronwall lemma (see [57], Part 1.6) guarantees the uniform boundedness both of ${\left\{u^m\right\}}_{m=1}^{\infty }$ in $\mathcal{V}$ and of ${\left\{{\tau }^m\right\}}_{m=1}^{\infty }$ in $\mathcal{H}$ for each $t\in \mathcal{I}$. The Eberlein–Shmul’yan theorem (see [62], Part 3.4) then guarantees, up to appropriate subsequences, the existence of weak limits of ${\left\{u^m\right\}}_{m=1}^{\infty }$ in $L^{\infty }(\mathcal{I},\mathcal{V})$ and of ${\left\{{\tau }^m\right\}}_{m=1}^{\infty }$ in $L^{\infty }(\mathcal{I},\mathcal{M})$, denoted by u and $\tau$. Coming back to (7) and (8), this enables us to make the final limit passage to (2) and (8). However, some more detailed convergence results can be derived, e.g., from the compact embeddings mentioned in the text block before (10). In particular, ${\left\{u^m\right\}}_{m=1}^{\infty }$ converges to u even strongly in $L^{\infty }(\mathcal{I},\mathcal{H})$; similar results can be repeated for ${\left\{\psi \left(u\right)\right\}}_{m=1}^{\infty }$ and $\psi \left(u\right)$, where $\psi$ is a continuous function.

3.3. The Second Step: Incorporation of Damage

We demonstrate this approach on our slightly upgraded model problem. We can multiply $\mathcal{C}$, as shown in the lower branch of Figure 1, by a certain scalar factor $1-\mathcal{D}\left(u\right)$, with positive irreversible real non-negative damage measure $\mathcal{D}\left(u\right)$ lesser than a certain constant ${\mathcal{D}}_{*}$ with values in $\mathcal{Z}$ between 0 and 1. In particular, ${\mathcal{D}}_{*}=0$ (no damage) is allowed; thus, we come back to our original model problem, but (at least) ${\mathcal{D}}_{*}<1$ is required everywhere to avoid the transfer up to stage d) without adequate physical and mathematical analysis. The inputs for evaluation of ${\mathcal{D}}_{*}$ should be some invariant strain or stress values, to preserve the objectivity of such an analysis.

More exactly, we can work with the five-step gradual evaluation of $\mathcal{D}$ here, as developed in [42]:

(i)

For any fixed $t\in \mathcal{I}$, compute three scalar principal values ${ϵ}_i$, $i\in \{1,2,3\}$, corresponding to $\epsilon \left(u\right)$, the computations being made on $\Omega$. Denoting by I the unit matrix in ${\mathbb{R}}^{3\times 3}$, it is necessary to find three eigenvalues ${ϵ}_i$ satisfying

$$det\left(\epsilon \left(u\right)-{ϵ}_i\left(u\right)I\right)=0.$$

(20)

(ii)

Evaluate an equivalent strain

$$\tilde{\epsilon }\left(u\right)=\omega (-{ϵ}_{1-}\left(u\right),{ϵ}_{1+}\left(u\right),-{ϵ}_{2-}\left(u\right),{ϵ}_{2+}\left(u\right),-{ϵ}_{3-}\left(u\right),{ϵ}_{3+}\left(u\right))$$

(21)

for ${ϵ}_{i+}\left(u\right)$ and ${ϵ}_{i-}\left(u\right)$ with $i\in \{1,2,3\}$ again, by ${ϵ}_{i+}\left(u\right)$ and ${ϵ}_{i-}\left(u\right)$ in (21) denoting the positive and negative parts of ${ϵ}_i\left(u\right)$ from (20). A bounded continuous function $\omega$ of six real non-negative arguments (at most three non-zero ones) must be prescribed; an instructive example of such a function $\omega$ can be found in [63].

(iii)

Introduce the nonlocal form of $\tilde{\epsilon }\left(u\right)$ from (21), motivated by [64], as

$$\overline{\epsilon }\left(u(x,t)\right)={\int }_{\Omega }\mathcal{K}(x,\tilde{x})\tilde{\epsilon }\left(u(\tilde{x},t)\right)\mathrm{d}\tilde{x},{\int }_{\Omega }\mathcal{K}(x,\tilde{x})\mathrm{d}\tilde{x}=1$$

(22)

for all $x\in \Omega$, with a fixed $t\in \mathcal{I}$. A regularising kernel $\mathcal{K}\in L^2(\Omega \times \Omega )$, as suggested by [65], is required in (22) as well.

The following approach avoids the problems connected with the non-existence of the solution in the sense of examples from [66,67]. Using the Dirac measure $\delta$, the seemingly simple choice $\mathcal{K}(x,\tilde{x})=\delta (x-\tilde{x})$, followed by $\overline{\epsilon }\left(u\right)=\tilde{\epsilon }\left(u\right)$, is not allowed, in contrast to the Gaussian kernel from [68] or to the various classes of radial basis function approximations from [69,70].

(iv)

Applying the result of (22), evaluate a trial value of damage

$$\widehat{\mathcal{D}}\left(u\right)=\varpi \left(\overline{\epsilon }\left(u\right)\right),$$

(23)

using a non-decreasing real continuous function $\varpi$.

(v)

Force the irreversibility of damage by (23), i.e., the non-decreasing damage factor in time, using the formula

$$\mathcal{D}\left(u(.,t)\right)=\underset{0\le \tilde{t}\le t}{max}\widehat{\mathcal{D}}\left(u(.,\tilde{t})\right).$$

(24)

A practical design of $\omega$, $\mathcal{K}$, and $\varpi$ may be difficult since any reasonable setting of material parameters obtained from experiments leads to a non-trivial sensitivity and inverse problems, which are frequently ill-conditioned. On the other hand, accenting the mathematical point of view presented in [58], Part 2.2, where two different versions of the proof are presented, entails the compactness of any operator from (22) acting over $L^2(\Omega )$.

Then, the generalized version of (4), involving certain damages, boils down to

$$\sigma =\tau +(1-\mathcal{D}(u\left)\right)\mathcal{C}\epsilon \left(u\right).$$

(25)

For sake of making the implementation of $\mathcal{D}\left(u\right)$ simple, brief, transparent and comparable with the preceding step, we do not change any of the other constitutive equations. From the point of view of the scheme from Figure 1, we modify only $\mathcal{C}$ in its lower branch by means of $(1-\mathcal{D}(u\left)\right)\mathcal{C}$. It is obvious that more advanced extensions of this scheme as well as the related constitutive equations may be useful in practical settings of real material parameters.

We venture to repeat all of the investigations from the preceding step, except the unchanged formulae, namely (2), (3), (6), (8), and (9). Instead of (5), we replace (4) by (25) and obtain

$$\left(\right(\epsilon \left(v\right),\tau \left)\right)+\left(\right(\epsilon \left(v\right),(1-\mathcal{D}(u\left)\right)\mathcal{C}\epsilon \left(u\right)\left)\right)=(v,f)+\langle v,g\rangle ,$$

(26)

Consequently, (7) must be replaced by

$$\left((\epsilon \left(v\right),{\tau }_s^m)\right)+\left((\epsilon \left(v\right),(1-\mathcal{D}\left(u\right))\mathcal{C}\epsilon \left(u_s^m\right))\right)=(v,f_s^m)+\langle v,g_s^m\rangle .$$

(27)

Although the theorem of Lax–Milgram cannot be directly applied at this point, there is fortunately another classical result regarding the solvability of (27) and (8) together with (6), which is available in [71]. Instead of (18), the sum of (27) and (9) with $v=u_s^m-u_{s-1}^m$ and $\gamma ={\tau }_s^m$ yields

$$\begin{array}{ccc}\hfill & \hfill & \hfill \left(({\tau }_r^m,{\mathcal{C}}^{-1}{\tau }_r^m/\alpha )\right)+h\sum _{s=1}^r\left(({\tau }_s^m-{\tau }_{s-1}^m,{\mathcal{C}}^{-1}({\tau }_s^m-{\tau }_{s-1}^m)\alpha )\right)\\ \hfill & \hfill & \hfill +2h\sum _{s=1}^r\left(({\tau }_s^m,{\mathcal{C}}^{-1}{\tau }_s^m/\beta )\right)+\left((\epsilon \left(u_r^m\right),\mathcal{C}\epsilon \left(u_r^m\right))\right)\\ \hfill & \hfill & \hfill +\sum _{s=1}^r\left((\epsilon (u_s^m-u_{s-1}^m),(1-\mathcal{D}\left(u_s\right))\mathcal{C}\epsilon \left(u_s^m\right))\right)\\ \hfill & \hfill & \hfill +\sum _{s=1}^r(\eta u_{s-1}^m+{\eta }^{-1}{f}_s^m,\eta u_{s-1}^m+{\eta }^{-1}{f}_s^m)\\ \hfill & \hfill & \hfill +\sum _{s=1}^r\langle \tilde{\eta }{u}_{s-1}^m+{\tilde{\eta }}^{-1}{g}_s^m,\tilde{\eta }{u}_{s-1}^m+{\tilde{\eta }}^{-1}{g}_s^m\rangle \\ \hfill & \hfill & \hfill =\eta h\sum _{s=1}^r(u_{s-1}^m,u_{s-1}^m)+\tilde{\eta }h\sum _{s=1}^r(u_{s-1}^m,u_{s-1}^m)\\ \hfill & \hfill & \hfill +{\eta }^{-1}h\sum _{s=1}^r(D{f}_s^m,D{f}_s^m)+{\tilde{\eta }}^{-1}h\sum _{s=1}^r\langle D{g}_s^m,D{g}_s^m\rangle .\end{array}$$

(28)

The details of the following approach can be omitted because only the fifth left-hand-side additive term of (28) needs to be handled in another way, analogous to the first step. Namely,

$$\begin{array}{ccc}\hfill & \hfill & \hfill 2\sum _{s=1}^r\left(\left(\epsilon (u_s^m-u_{s-1}^m)(1-\mathcal{D}\left(u_s\right))\mathcal{C}\epsilon \left(u_s^m\right)\right)\right)\\ \hfill & \hfill & \hfill =\sum _{s=1}^r\left((\epsilon \left(u_s^m\right),(1-\mathcal{D}\left(u_s\right))\mathcal{C}\epsilon \left(u_s^m\right))\right)\\ \hfill & \hfill & \hfill -\sum _{s=1}^r\left((\epsilon \left(u_{s-1}^m\right),(1-\mathcal{D}\left(u_{s-1}\right))\mathcal{C}\epsilon \left(u_{s-1}^m\right))\right)\\ \hfill & \hfill & \hfill +\sum _{s=1}^r\left(\left(\epsilon (u_s^m-u_{s-1}^m)(1-\mathcal{D}\left(u_s\right))\mathcal{C}\epsilon (u_s^m-u_{s-1}^m)\right)\right)\\ \hfill & \hfill & \hfill +\sum _{s=1}^r\left(\left(\epsilon \left(u_{s-1}^m\right)(\mathcal{D}\left(u_s^m\right)-\mathcal{D}\left(u_{s-1}^m\right))\mathcal{C}\epsilon \left(u_{s-1}^m\right)\right)\right)\\ \hfill & \hfill & \hfill =\left((\epsilon \left(u_r^m\right),(1-\mathcal{D}\left(u_r\right))\mathcal{C}\epsilon \left(u_r^m\right))\right)\\ \hfill & \hfill & \hfill +\sum _{s=1}^r\left(\left(\epsilon (u_s^m-u_{s-1}^m)(1-\mathcal{D}\left(u_s\right))\mathcal{C}\epsilon (u_s^m-u_{s-1}^m)\right)\right)\\ \hfill & \hfill & \hfill +\sum _{s=1}^r\left(\left(\epsilon \left(u_{s-1}^m\right)(\mathcal{D}\left(u_s^m\right)-\mathcal{D}\left(u_{s-1}^m\right))\mathcal{C}\epsilon \left(u_{s-1}^m\right)\right)\right).\end{array}$$

(29)

Since $\mathcal{D}\left(u_s^m\right)\ge 1-{\mathcal{D}}_{*}>0$ holds everywhere for any $s\in \{1,\dots ,r\}$, only the last right-hand-side additive term of (29) deserves a special mention. Clearly, $\mathcal{D}\left(u_s^m\right)-\mathcal{D}\left(u_{s-1}^m\right)$ is always non-negative due to (24) formally (from the mechanical point of view, no damage recovery is allowed); thus, this term can be estimated by zero from below. The modification of all remaining items of the first step is then straightforward, taking the strong convergence ${\left\{\mathcal{D}\left(u^m\right)\right\}}_{m=1}^{\infty }$ to $\mathcal{D}\left(u\right)$ into account.

4. Possible Generalizations

A curious reader can see that some parts of the derivation of existence and convergence results in both steps of a model problem can be performed in a more general form, admitting less transparent assumptions and more complicated proofs. In particular, we have $f\in W^{1,2}(\mathcal{I},\mathcal{H})$ and $g\in W^{1,2}(\mathcal{I},\mathcal{G})$, but no explicit result for any derivative of $u\in L^2(\mathcal{I},\mathcal{V})$. Differentiating (3) with respect to $t\in \mathcal{I}$, one can expect, due to its linearity, despite the presence of a nonlinear damage factor $\mathcal{D}$, to find the first derivative of u (at least) in $L^2(\mathcal{I},{\mathcal{V}}^{*})$; ${\mathcal{V}}^{*}$ here denotes the dual space to $\mathcal{V}$. Another example of such an argument can be seen in (ii) preceding (10) and (11); the compact embedding of $\mathcal{H}$ into $\mathcal{V}$ is considered, together with the continuous embedding of $\mathcal{G}$ into $\mathcal{V}$, but even the compact embedding of $L^{6-\xi }{(\Omega )}^3$ into $\mathcal{V}$ with $0<\xi \le 4$, as well as the compact embedding of $L^{4-\xi }{(\partial \Omega )}^3$ into $\mathcal{V}$ with $0<\xi \le 2$, can be guaranteed.

Unfortunately, the exploitation of such formal results on function spaces in our considerations is not so easy. (i) Most mathematical proofs with such non-Hilbert (but still reflexive Banach) spaces are more complicated, with fewer available standard arguments from linear functional analysis. (ii) Most computational algorithms suffer from the absence of orthogonal projections from certain original infinite-dimensional to approximating finite-dimensional ones, which can be performed just for Hilbert spaces with the help of scalar products between their elements. Some selected results remain valid for a more general class of open sets than for a class of domains with Lipschitzian boundaries, too. However, we shall sketch some deeper generalizations here now, whose criteria of relevance come from their expected contribution to a substantial upgrade of modelling and simulation techniques for quasi-brittle composites.

4.1. From Quasi-Static to Dynamic Models

Traditional numerical analyses of development (or stagnation) of both macro- and microscopic damage in (not only) composites rely on some quasi-static formulation, comparable with our model problem. Some of them do not even introduce any finite time interval, which is replaced by a certain study of the transfer between static configurations. However, such an approach is not realistic for quick damage processes, such as those analysing the seismic effects on the stability of engineering structures or the mutual contacts and/or impacts of deformable bodies. Some existence and convergence properties for such a problem, whose damage is added to the simple Kelvin parallel viscoelastic model, are studied in [39]; for the incorporation of selected methods from the graph theory for effective searches of candidates to multiple contact surfaces, cf. [41].

Following our model problem, its simplest extension works with an inertia force $\rho \ddot{u}$, where a dot symbol is used instead of $\partial /\partial t$ on $\Omega$ and $\rho \in \mathcal{Z}$ means the material density. Thus, the modification of (2) can be

$$\left((\epsilon \left(v\right),\sigma )\right)=(v,f-\rho \ddot{u})+\langle v,g\rangle .$$

(30)

One can then search, still taking $\tau \in L^2(\mathcal{I},\mathcal{M})$ into account, for $u\in L^2(\mathcal{I},\mathcal{V})$ with $\dot{u}\in L^2(\mathcal{I},\mathcal{H})$ and $\ddot{u}\in L^2(\mathcal{I},\mathcal{V})$; these requirements may be modified slightly due to yet another additional term to $\rho \ddot{u}$ for some estimate of dissipative effects, depending on $\ddot{u}$ typically, or even on some fractional derivative by t, i.e., for the transfer of mechanical energy to thermal energy, lost in the environment. In addition to the Rothe sequences of simple abstract functions, as introduced in our considerations everywhere, one also needs another type of the Rothe sequences of linear Lagrange splines to cover the first-time derivatives and certain additional estimates in dual spaces to cover the second derivatives. Such an approach enables us to weaken the assumptions on the smoothness of f and g substantially; the requirement on the non-zero Hausdorff measure of $\Theta$ on $\partial \Omega$ can be removed in a proper dynamic formulation.

4.2. Macroscopic Cracks and Cohesive Interfaces

Up to now, we have not overcome stage (a), as introduced in Section 2, apart from the rather long history of attempts to also cover (at least) (b). Such phenomena as (i) the evolution of cracks, (ii) the propagation of dislocations, (iii) the diffusion on grain boundaries and (iv) the development of phase boundaries have been discussed in [72], modifying the standard finite element techniques by [73]. Research progress in the next decade is documented in [74], whereas [75,76] represent the recent regularisation approaches and search for reasonable, well-conditioned formulations, at least for some simplified problems, addressed to the interface between (b) and (c).

The so-called extended, generalised, and similar finite element formulations seem to be related just to the discretisation of $\Omega$, $\Theta$, and $\Gamma$, and moreover, to a certain set $\Lambda$ of potential candidates to internal interfaces, whose behaviour, characterised by opening and closing (changing a tight contact to a free surface on $\Lambda$ locally), driven by various types of loads, can be determined by the so-called traction separation laws by [37,77]. This means that for certain scalar tractions $\mathfrak{T}\left(\right[u\left]\right)$, introduced on $\Lambda$ locally, together with their nonlinear continuous dependence on $\left[u\right]$, where $[.]$ must be read as jumps in values (in the sense of traces again), one can only join one more additive term, similar to $\langle v,g\rangle$, to the left-hand side of (2) or (30) in the form ${\langle \left[v\right],\mathfrak{T}\left(\left[u\right]\right)\rangle }_{\Lambda }$; ${\langle .,.\rangle }_{\Lambda }$ here highlights the integration over $\Lambda$ instead of $\Gamma$. Thus, we obtain an extended version of (30):

$$\left((\epsilon \left(v\right),\sigma )\right)+{\langle \left[v\right],\mathfrak{T}\left(\left[u\right]\right)\rangle }_{\Lambda }=(v,f-\rho \ddot{u})+\langle v,g\rangle .$$

(31)

From the point of view of finite element algorithms, some special types of basis functions are implemented on $\Lambda$ in (31), specific for crack tips, their open and closed parts, etc., respecting three basic traditionally distinguished modes: I opening, II shearing, and III tearing, due to the orientation of forces relative to the crack.

Nevertheless, this works, with respect to an additional source of nonlinearity in $\mathfrak{T}\left(\right[.\left]\right)$, for rather simple techniques with cracks allowed on (at most) a finite number of a priori prescribed interfaces such that $\Omega$ can be compounded from a finite number of Lipschitzian domains. In the opposite case, e.g., in newer improvements of extended finite element techniques, new cracks can initiate in directions and locations unexpected originally, which, from the point of view of mathematical analysis, leaves the theory of function spaces on Lipschitzian domains; the difficulties of such generalisation (with still unclosed problems) can be seen from [78,79]. However, this should be handled properly, joining smeared and macroscopic approaches in advanced coupled failure engineering models like [47,80]. Let us notice a regularisation compromise of [81] for large distributed and parallel computations, too.

4.3. Proper Thermodynamic Formulations

However, real-world phenomena do not respect engineering schemes like that of Figure 1 at a high-precision level needed for reliable modelling of (c) and (d)—at least partially, neglecting uncertainty of both loads and material parameters, including structural imperfections, with missing or incomplete data from the macroscopic point of view. The proper thermodynamic formulations of [82,83,84] work with the specific Gibbs energy and with the complementary form of the dissipation potential, where the material damage is described by a certain separate damage tensor from $\mathcal{E}$. Namely, the specific Gibbs energy can be seen as a function of four state variables, namely temperature, stress, damage and a set of internal parameters, corresponding to four dissipative variables, both included in the formulation of a dissipative potential.

Up to now, a complete physical and mathematical theory is not available. Sufficient smoothness of a dissipative potential is not guaranteed to admit its partial differentiation in a reasonable sense; thus, the theory of subdifferentials by [85] must be exploited. Practical computational attempts still consider the small strain theory with the fixed initial reference configuration and work with some simplified additive decomposition of the specific Gibbs energy with respect to particular state variables, which limits their application to (c) and beyond (c) substantially. However, the basic ideas of the computational model or elastic, viscous, and plastic deformation, including damage, can be inspired by [86], Chap. 8.1, with necessary modifications forced by the nonlinear analysis of composed structured deformations $(\kappa ,\mathfrak{G},G)$; this will be sketched in the final part of Section 4.4.

4.4. Structured Deformation

To overcome selected drawbacks of small strain formulations, some changes or time-dependent updates of a coordinate system may be helpful. This covers various adaptive remeshing techniques by [87], an updated Lagrangian method by [88], as well as an ALE (arbitrary Lagrangian Eulerian) formulation by [89]. However, a more precise geometrical description of deformation, with the support of scale bridging, is needed, too.

Such geometrical theory of structured deformation comes from [90], supplied by necessary physical consideration in [91]; certain results on energetic relaxation are presented in [92]. No small strain assumptions or additive decompositions are needed; theorems on compositions of sliding, mixing, etc., deformations are derived. Apart from the absence of effective computational algorithms applicable to real physical and engineering problems, the significant progress in the mathematical theory in this promising research area in the last decade is documented in [93,94,95,96,97].

The geometrical effects at the macroscopic level of submicroscopic disarrangements are captured by sequences of approximating, piecewise smooth deformations ${\mathfrak{F}}_n$ with $n\in \{1,2,\dots \}$ that converge (in some reasonable sense) to the macroscopic deformation field $\mathfrak{G}$ and whose gradients $\nabla {\mathfrak{F}}_n$ converge to a field G, allowed to differ from $\nabla \mathfrak{G}$. A structured deformation can be then introduced as a triple $(\kappa ,\mathfrak{G},G)$, where (i) $\kappa$ is a surface-like subset of $\Omega$, which represents a bounded open subset of the N-dimensional Euclidean space ${\mathcal{R}}^N$, (ii) $\mathfrak{G}:\Omega \to {\mathcal{R}}^N$ is an injective and piecewise differentiable map, and (iii) $G:\Omega \to {\mathcal{R}}^N$ is a piecewise continuous tensor field such that $c<detG\left(x\right)\le det\nabla \mathfrak{G}\left(x\right)$ at each point $x\in \Omega$, c being some positive constant independent of x. In view of this definition, (i) $\kappa$ describes pre-existing unopened macroscopic cracks; (ii) $\mathfrak{G}$ and $\nabla \mathfrak{G}$ describe macroscopic changes in the geometry of the body $\Omega$; (iii) G must be interpreted by the approximation theorem of [90]: There exists a sequence of injective piecewise smooth deformations ${\mathfrak{F}}_n$ and a sequence of surface-like subsets ${\kappa }_n$ such that, in the sense of $L^{\infty }$-convergence (using the standard notation of Lebesgue spaces everywhere), as limits of so-called simple deformations

$$\underset{n\to \infty }{lim}{\mathfrak{F}}_n,G=\underset{n\to \infty }{lim}\nabla {\mathfrak{F}}_n,\kappa =\bigcup _{n=1}^{\infty }\bigcap _{j=n}^{\infty }{\kappa }_j.$$

(32)

$M=\nabla \mathfrak{G}-G$ on $\Omega \setminus \kappa$ in (32) is interpreted as the Burgers microfracture tensor, related to the well-known Burgers vector from continuum theories of dislocations, whereas the multiplicative decompositions $\nabla \mathfrak{G}=M_{l}G=G{M}_r$ with the left and right microfracture tensors $M_l$, $M_r$ are useful for coupling of various types of structured deformations.

A more precise analysis needs to introduce the space of special functions of bounded variations, denoted by $SBV(\Omega ,{\mathcal{R}}^N)$, as a set or ${\mathcal{R}}^N$-valued functions with bounded variations

$$BV(\Omega ,{\mathcal{R}}^N)=\{\mathfrak{U}\in L^1(\Omega ,{\mathcal{R}}^N):D\mathfrak{U}\in \mathcal{M}(\Omega ,{\mathcal{R}}^{N\times N})\},$$

(33)

with D referring to distributional derivatives and $\mathcal{M}$ to Radon measures, such that the singular part of $D\mathfrak{U}$ in (33) is reduced to the jump part, i.e., the Cantor part of $D\mathfrak{U}$ in the sense of [98] vanishes. This makes it possible to introduce the space of structured deformation

$$SD(\Omega )=\{(\mathfrak{G},G):\mathfrak{G}\in SBV(\Omega ,{\mathcal{R}}^N),G\in L^1(\Omega ,{\mathcal{R}}^{N\times N})\};$$

(34)

$\nabla \mathfrak{G}-G$ derived from (34) refers to the disarrangement tensor. The following convergence result is available. For any $(\mathfrak{G},G)\in SD(\Omega )$, there exists such ${\mathfrak{U}}_n\in SBV(\Omega ,{\mathcal{R}}^N)$ that

$${\mathfrak{U}}_n\to \mathfrak{G}\mathrm{in}{L}^1(\Omega ,{\mathcal{R}}^N),\nabla {\mathfrak{U}}_n\stackrel{*}{\rightharpoonup }G\mathrm{in}\mathcal{M}(\Omega ,{\mathcal{R}}^{N\times N}),$$

(35)

using the standard notion of strong- and weak-star convergence. The original proof of this result by [99] combines the Alberti theorem by [100] with certain approximations of functions with bounded variations by piecewise constant ones; an alternative proof using a weaker approximation (avoiding [100]) can be found in [94].

The jumps in ${\mathfrak{F}}_n$ contribute to the interfacial part of an initial energy response $E\left({\mathfrak{F}}_n\right)$. They diffuse throughout portions of the body and generate, in the limit, all bulk and interfacial parts of a relaxed energy response $I(\mathfrak{G},G)$. Introducing bulk densities W and interfacial densities $\Psi$, a $(N-1)$-rectifiable jump set $S\left(u\right)$ with its unit normals $\nu \left(\mathfrak{U}\right)$ for any admissible function u with non-zero jumps $\left[\mathfrak{U}\right]$ there, together with the functional

$$E\left(\mathfrak{U}\right)={\int }_{\Omega }W(\nabla \mathfrak{U}\left(x\right))\mathrm{d}x+{\int }_{S\left(\mathfrak{U}\right)}\Psi (\left[\mathfrak{U}\left(x\right)\right],\nu \left(\mathfrak{U}\right))\mathrm{d}s\left(x\right).$$

(36)

The aim is to come from (36) to

$$I(\mathfrak{G},G)=inf\left(\underset{n\to \infty }{lim\; inf}E\left({\mathfrak{U}}_n\right)\right)$$

(37)

with ${\mathfrak{U}}_n\in SBV(\Omega ,{\mathcal{R}}^N)$ satisfying (35). For a rather wide class of energy functionals with carefully introduced W and $\Psi$, ref. [99] performs such relaxation utilising the so-called blow-up method by [101]. For more details on explicit formulae for interfacial energies, optimal design, and dimension reduction, see [93].

Now, we are able to come back to the considerations of Section 4.3, with $\theta$ referring to the positive Kelvin temperature. Let us use the symbol ${\partial }_z$ for partial derivatives with respect to a variable z here, as well as upper dots for partial time derivatives. Taking, instead of $I(\mathfrak{G},G)$ by (37), some $\psi (F,G,\mathcal{Q},\theta )$, interpretable as the Helmholtz free energy, containing the deformation gradient $F=\nabla g$, admitting needed multiplicative decompositions, one can formulate the internal energy

$$\omega (F,G,\mathcal{Q},\theta ,\mathfrak{H})=\psi (F,G,\mathcal{Q},\theta )+\mathfrak{H}\theta ,$$

(38)

whose setting

$$\mathfrak{H}=-{\partial }_{\theta }\psi (F,G,\mathcal{Q},\theta )$$

(39)

for the specific entropy $\mathfrak{H}$ is well known as the Gibbs relation; $\mathcal{Q}$ denotes a vector of appropriate internal parameters, and $\zeta (\dot{F},\dot{G},\dot{\mathcal{Q}})$ will refer to certain potential of dissipative forces. Differentiating (38) in time, thanks to the obvious consequence of (39) ${\partial }_{\theta }\psi \dot{\theta }+\dot{\theta }\mathfrak{H}=0$, we obtain

$$\dot{\omega }={\partial }_F\psi :\dot{F}+{\partial }_G\psi :\dot{G}+{\partial }_{\mathcal{Q}}\psi ·\dot{\mathcal{Q}}+\theta \dot{\mathfrak{H}}.$$

(40)

The derivation of the system of partial differential equations of evolution, coming from (40), comparable with (31), is rather difficult, with expectable complicated results, even in special cases significant in engineering practice; for more details, see [48].

4.5. Contribution of Differential Geometry

Another research challenge of continuum mechanics is overcoming the physical notion of Boltzmann continuum, utilised in all of our exact considerations except the theory of structured deformation, which is open to more general conceptions. This idea dates back to the beginning of the 20th century when the so-called Cosserat continuum was introduced in [102]. This concept can be considered as a starting point for building new theories, developed for the modern disciplines like linearised continuum mechanics, overviewed by [103], which is tailored for various engineering methods using Cosserat continuum; see, e.g., [104,105], with an available advanced surface parametrisation from [106].

Among the other modelling methods in continuum mechanics, there are the methods of differential geometry. The endeavours in this field are represented by [107,108,109], but there are also historical papers, namely [110], introducing the concept of the holonomic, non-holonomic, and semi-holonomic jet together with the mechanical motivations. Nevertheless, the Cosserat medium can still be considered as the starting and motivation point for the development of geometric methods under discussion. Further, there are papers and books (see [111,112]) devoted to giving the mechanical content to fundamental geometrical concepts and theories studied systematically in the classical geometrical monographs, e.g., [113]. Nevertheless, the monograph [109] above is crucial and is devoted to building the continuum mechanics theories focused on homogeneity, inhomogeneities, and the evolution processes from the point of view of modern differential geometry, namely the theory of jets, bundles, and connections.

Apart from the elementary mechanical concept of a configuration as a counterpart of a local map on a three-dimensional manifold followed by the fundamental concepts of frame and frame bundle, there are the crucial concepts of material isomorphism, symmetry group, archetype, uniformity field, material G-structures, material grupoid, and in the centre, the concept of homogeneity. In the first-order elastic materials, all of the concepts are treated in terms of the geometrical concepts of the frame bundle, principal bundle, principal and linear connection, together with the related tensors like the torsion and the curvature.

In Ref. [109], the so-called second-order uniform materials are studied followed by their modification to functionally graded materials. Configurations are identified with the choice of the second-order material archetype and the elements with the second-order jets of material implants. As for the homogeneity in such spaces, the situation is more complicated in comparison with the first-order materials modelling the so-called microlinear structure, since the homogeneity can be either considered in the sense of the existence of a configuration with translations acting as material isomorphism or in the weaker sense of the homogeneity defined with respect to a given archetype. In any case, it is necessary to work with higher-order frame bundles and principal connections and also with the associated linear connections. In Ref. [109], Cosserat media are also studied. The Cosserat media approach handles, e.g., cementitious composites including concrete. The so-called macromedium or matrix has its geometrical counterpart in the basis of the principal bundle, while its elements determine the events within the so-called grains, the fibres of the principal bundle obtained as a reduction of the bundle of non-holonomic frames with respect to the subgroup of a jet group.

The efforts for generalising both of the basic models open the door to the application of Weil bundle techniques and the theory of homogeneous spaces, namely generalised Stiefel and Grassmann manifolds; see [43,44,114,115] in a more wide context. Applying the Weil approach will enable not only the investigation of layers in bodies, but also the incorporation of time and the consequent evolution.

5. Computational Approaches

To demonstrate available computational approaches, let us come back to our model problem, formulated in Section 3, in the time-discretised form, but still in infinite-dimensional spaces $\mathcal{G}$ and $\mathcal{V}$. We shall work with usual notations from the standard finite element method, compatible with [73], for simplicity.

5.1. Finite-Dimensional Schemes

Let ${\mathcal{G}}^H$ and ${\mathcal{V}}^H$ be some finite-dimensional approximation of $\mathcal{G}$ and $\mathcal{V}$; in some particular cases (if possible), ${\mathcal{G}}^H$ and ${\mathcal{V}}^H$ are allowed to be certain subspaces of ${\mathcal{G}}^H$ and ${\mathcal{V}}^H$ directly. Some regular decomposition of $\Omega$ and its boundary to finite elements is considered; an upper index H refers to a reference mesh element edge length. Consequently, the analysis of limit passage $H\to 0$ shows the quality of convergence for a selected problem, i.e., for a modification of (9) in the form

$$\left(({\gamma }^H,{\mathcal{C}}^{-1}({\tau }_s^{mH}-{\tau }_{s-1}^{mH})/\alpha )\right)+h\left(({\gamma }^H,{\mathcal{C}}^{-1}{\tau }_s^{mH}/\beta )\right)=\left(({\gamma }^H,\epsilon (u_s^{mH}-u_{s-1}^{mH}))\right).$$

(41)

for any ${\gamma }^H\in {\mathcal{G}}^h$ and that of (27) in the form

$$\left((\epsilon \left(v^H\right),{\tau }_s^{mH})\right)+\left((\epsilon \left(v^H\right),(1-\mathcal{D}\left(u_{s*}^{mh}\right))\mathcal{C}\epsilon \left(u_s^{mH}\right))\right)=(v^H,f_s^m)+\langle v^H,g_s^m\rangle .$$

(42)

for any $v^H\in {\mathcal{V}}^H$. This leads to a sparse system of nonlinear equations, which may force an additional iteration procedure inside a time step $s\in \{1,\dots ,m\}$ with (at most) linear convergence, as usual in methods of successive approximation. However, the original choice requires the exploitation of an appropriate algorithm for the analysis of nonlinear equations, such as the inexact Newton method (see [116]), the conjugate gradient method (see [117]), or the Nelder–Mead nonlinear simplex method (see [118]), with certain risks of non-convergence or practical stagnation; the optimal selection should be performed with respect to the specific form of $\mathcal{D}(.)$.

All preceding comments are valid for our model problem from Section 3 without any modification: in the first step, where $\mathcal{D}(.)=0$, even in the simplified form, because (27) generates a system of linear algebraic equations directly. The same approach can be applied in Section 4.1 too, with an obvious upgrade. An optimal setting of some extended finite element technique for Section 4.2 is more sophisticated. The global form of (41) and (42) can be preserved, but the practical computational algorithm may depend on the selection of adaptive enrichment functions for ${\mathcal{G}}^H$ and ${\mathcal{V}}^H$, cf. [119,120]. The design and analysis of numerical procedures for Section 4.3, Section 4.4 and Section 4.5 contains numerous open questions, deserving deep particular studies in each of these cases.

5.2. Illustrative Example

The practical validation of the theoretical results deduced above has been performed in the form of a simulation of the four-point bending test of a wooden beam, as shown in Figure 2. The software implementation of direct computations on the basis of the software package RFEM 6 is a result of a cooperation of several members of the research team at BUT with the companies FEM Consulting (Brno, Czech Republic) and Dlubal Software (Tiefenbach, Germany). The inverse problems, beyond the scope of this article, have been solved by means of the in-house software utilising particle swarm optimisation. Its aim is to identify optimal parameters of the SLS model and also generalised Kelvin chains with more components, corresponding to well-designed laboratory experiments. Much more information about these non-trivial problems can be found in [46].

More complex results are now available at BUT for various types of concrete and composite material samples and structures as outputs from the grant project of the Technological Agency of the Czech Republic Advanced Software Tool for Concrete and Composite Structures Analysis during Construction Stages with Respect to Rheology (M. Trcala, BUT, 2021–24). Their evaluation for further publication is currently in progress.

6. Conclusions

The aim of this article, based on the research experience of both authors and their collaborators at BUT, was (i) conducting a thorough overview of recent results on mathematical and computational aspects of modelling and simulation of quasi-brittle composites under mechanical loads and (ii) warning against a gap between theoretical mathematics and engineering computations and suggesting some ways to fill it in. A sufficiently simple but still interesting model problem was considered of a viscoelastic deformable body with potential damage. This enables opening all considerations of the article, including the new existence and convergence results, to a wide class of readers, not only to the limited community of experts in advanced functional analysis.

The research challenge for the near future can be seen in the passage from the quasilinear considerations, presented in Section 3, to the fully nonlinear ones, sketched briefly in Section 4, covering finite strain dynamical phenomena and further physical influences. In the first step, such passage could still rely on the generalised Kelvin chains and some implementation of a coupled micro- and macroscopic damage model. This must contain both formal mathematical verification and practical validation of results, including sufficiently simple setting of material parameters, valid both under laboratory conditions at BUT and during in situ observations.