Fatigue Effects in Elastic Materials with Variational Damage Models: A Vanishing Viscosity Approach

Abstract

We study the existence of quasistatic evolutions for a family of gradient damage models which take into account fatigue, that is the process of weakening in a material due to repeated applied loads. The main feature of these models is the fact that damage is favoured in regions where the cumulation of the elastic strain (or other relevant variables, depending on the model) is higher. To prove the existence of a quasistatic evolution, we follow a vanishing viscosity approach based on two steps: we first let the time step $\tau$ of the time discretisation and later the viscosity parameter $\varepsilon$ go to zero. As $\tau \rightarrow 0$, we find $\varepsilon$-approximate viscous evolutions; then, as $\varepsilon \rightarrow 0$, we find a rescaled approximate evolution satisfying an energy-dissipation balance.

A Auxiliary Results

1.1 The Essential Variation

In this appendix, X denotes a measure space. We do not label the measure on X and the notions of $L^p$ space and of a.e. equivalence refer to the measure on X. Moreover, we fix $n \ge 1$.

We define here the notion of essential variation, namely the variation for a time-dependent family of measurable functions, in the sense of a.e. inequality.

Definition A.1

Let us consider a function $t \mapsto \zeta (t)$, with $\zeta (t) :X \rightarrow \mathbb {R}^n$. Let $0 \le s \le t \le T$. The essential variation of $\zeta$ in the interval [s, t] is the function ${{\mathrm{ess\,Var}}}(\zeta ;s,t) :X \rightarrow [0,+\infty ]$ defined by

$$\begin{aligned} {{\mathrm{ess\,Var}}}(\zeta ;s,t) := \mathop {{{\mathrm{ess\,sup}}}}\limits _{s=s_0< \dots < s_m = T} \Big \{ \sum _{j=0}^m |\zeta (s_j) - \zeta (s_{j-1})| \Big \}, \end{aligned}$$

where the essential supremum is taken over all partitions $0=s_0< \dots < s_m = t$, $m \in \mathbb {N}$.

Remark A.2

For every $t_1 \le t_2 \le t_3$, we have

$$\begin{aligned} {{\mathrm{ess\,Var}}}(\zeta ;t_1,t_3) = {{\mathrm{ess\,Var}}}(\zeta ;t_1,t_2) + {{\mathrm{ess\,Var}}}(\zeta ;t_2,t_3) \, \quad \text {a.e. in } X. \end{aligned}$$

For completeness, we recall here the definition of the essential supremum of a family of measurable functions.

Definition A.3

Let $(v_i)_{i\in I}$ be a family of measurable functions from X to $[-\infty ,\infty ]$. Let ${\overline{v}} :X \rightarrow [-\infty ,\infty ]$ be a measurable function such that

• (i) ${\overline{v}} \ge v_i$ a.e. in X, for every $i\in I$;

• (ii) if $v :X \rightarrow [-\infty ,\infty ]$ is a measurable function such that $v \ge v_i$ a.e. in X, for every $i\in I$, then $v \ge {\overline{v}}$ a.e. in X.

The functions ${\overline{v}}$ is called an essential supremum of the family $(v_i)_{i\in I}$. In fact, there exists a unique (up to a.e. equivalence) essential supremum ${\overline{v}}$ of the family $(v_i)_{i \in I}$. We denote it by $\displaystyle {{\mathrm{ess\,sup}}}_{i \in I} v_i := \overline{v}$.

In the next proposition, we provide an explicit formula for the essential variation of a function $\zeta$ that is absolutely continuous in time. A quick survey about the notion and the main properties of the Bochner integral can be found in appendix of Brezis (1973); for a more detailed treatment of the subject, we refer to Dunford and Schwartz (1988).

Proposition A.4

Let $p \in [1, \infty )$, and let $\zeta \in AC([0,T];L^p(X;\mathbb {R}^n))$. Then

$$\begin{aligned} {{\mathrm{ess\,Var}}}(\zeta ;0,t)(x) = \int _0^t |{\dot{\zeta }}(r;x) | {\, \mathrm {d} r}, \quad \text {for a.e. } x \in X , \end{aligned}$$

where the integral in the right-hand side is a Bochner integral in $L^p(X)$.

Proof

We start by claiming that

$$\begin{aligned} {{\mathrm{ess\,Var}}}(\zeta ;0,\cdot ) \in AC([0,T];L^p(X)) \quad \text {and} \quad \frac{\, \mathrm {d}}{{\, \mathrm {d} t}} {{\mathrm{ess\,Var}}}(\zeta ;0,t) = |{\dot{\zeta }}(t)| \text { in } L^p(X). \end{aligned}$$

(A.1)

To prove the claim, let us fix $s \le t$ and a partition $s=s_0< \dots < s_m = t$. By the absolute continuity of $\zeta$, we obtain that

$$\begin{aligned} \sum _{j=1}^m |\zeta (s_j) - \zeta (s_{j-1})| = \sum _{j=1}^m \Big | \int _{s_{j-1}}^{s_j} {\dot{\zeta }}(r) \, {\, \mathrm {d} r} \Big | \le \sum _{j=1}^m \int _{s_{j-1}}^{s_j} | {\dot{\zeta }}(r) | \, {\, \mathrm {d} r} = \int _{s}^{t} | {\dot{\zeta }}(r) | \, {\, \mathrm {d} r} \, \end{aligned}$$

(A.2)

a.e. in X, where the last integral is a Bochner integral in $L^p(X)$. Note that the second inequality in (A.2) can be proved, e.g., with an approximation argument via step functions. Taking the essential supremum in (A.2), by Remark A.2 we deduce that

$$\begin{aligned} {{\mathrm{ess\,Var}}}(\zeta ;0,t) - {{\mathrm{ess\,Var}}}(\zeta ;0,s)\le \int _{s}^{t} | {\dot{\zeta }}(r) | \, {\, \mathrm {d} r} \quad \text { a.e. in } X. \end{aligned}$$

(A.3)

Inequality (A.3) computed for $s = 0$ yields, in particular, that ${{\mathrm{ess\,Var}}}(\zeta ; 0, t) \in L^p(X)$ for every $t \in [0,T]$. Moreover, it shows that ${{\mathrm{ess\,Var}}}(\zeta ;0,\cdot ) \in AC([0,T];L^p(X))$. By (A.3) and by Lebesgue’s differentiation theorem for vector-valued functions (Dunford and Schwartz 1988, p. 217), we get that

$$\begin{aligned} \frac{\, \mathrm {d}}{{\, \mathrm {d} t}} {{\mathrm{ess\,Var}}}(\zeta ;0,t) = \lim _{s\rightarrow t^-} \frac{{{\mathrm{ess\,Var}}}(\zeta ;0,t) - {{\mathrm{ess\,Var}}}(\zeta ;0,s)}{t-s} \le | {\dot{\zeta }}(t) | \end{aligned}$$

if t is a differentiability point for ${{\mathrm{ess\,Var}}}(\zeta ;0,\cdot )$ and it is a Lebesgue point for $|{\dot{\zeta }}|$, the limit being taken with respect to the $L^p$-norm.

On the other hand, $s < t$ is a particular partition of the interval [s, t]; therefore,

$$\begin{aligned} \frac{|\zeta (t) - \zeta (s)|}{t-s} \le \frac{{{\mathrm{ess\,Var}}}(\zeta ; 0, t) - {{\mathrm{ess\,Var}}}(\zeta ; 0, s)}{t-s} \, \quad \text {a.e. in } X. \end{aligned}$$

Taking the limit as $s \rightarrow t^-$ with respect to the $L^p$-norm of both sides, we obtain

$$\begin{aligned} | {\dot{\zeta }}(t) | \le \frac{\, \mathrm {d}}{{\, \mathrm {d} t}} {{\mathrm{ess\,Var}}}(\zeta ;0,t) \, \quad \text {a.e. in } X, \end{aligned}$$

if t is a differentiability point for ${{\mathrm{ess\,Var}}}(\zeta ;0,\cdot )$ and $\zeta$. This proves that $\frac{\, \mathrm {d}}{{\, \mathrm {d} t}} {{\mathrm{ess\,Var}}}(\zeta ;0,t) = | {\dot{\zeta }}(t) |$.

Finally, since ${{\mathrm{ess\,Var}}}(\zeta ;0,\cdot ) \in AC([0,T];L^p(X))$, we conclude that

$$\begin{aligned} {{\mathrm{ess\,Var}}}(\zeta ;0,t) = \int _0^t \frac{\, \mathrm {d}}{{\, \mathrm {d} t}} {{\mathrm{ess\,Var}}}(\zeta ;0,r) \, {\, \mathrm {d} r} = \int _0^t | {\dot{\zeta }}(t) | \, {\, \mathrm {d} r} \, \quad \text {a.e. in } X. \end{aligned}$$

$\square$