1. Barrierefreie Word Dokumente und einfache PDfs

4 Minuten

<div style="position: relative; margin: 20px 20px 20px 20px; padding: 30px; padding-top:0;">
Hereditary Integral Formulation

\(\bullet\) Model assumption:


Linear superposition e.g. of small jumps in stresses


\(\bullet\) Initial conditions:


<center>\(t = 0\) \(\leadsto\) \(\sigma_0\) \(\leadsto\) \(\varepsilon = \sigma_0 \cdot J(t)\)</center>


\(\bullet\) Inkremental jump of stress:


<center>\(t = \tau\) \(\rightarrow\) \(\Delta \sigma\)</center>


\(\bullet\) Material response:


<center>\(\varepsilon(t) = \sigma_0 \cdot J(t) + \Delta \sigma \cdot J(t - \tau)\)</center>


\(\bullet\) Transition to continuous process:


<center>\(\varepsilon(t) = \sigma_0 \cdot J(t) + \int_{0}^{t}{J(t - \tau) d \sigma(\tau)}\)</center>

\(= \sigma_0 \cdot J(t) + \int_{0}^{t}{J(t - \tau) \frac{d\sigma}{d\tau} d \tau}\)


\(\bullet\) And after integration by parts:


<center>\(\varepsilon(t) = \sigma_0 \cdot J(t) + \left[J(t - \tau) \cdot \sigma(\tau) \right]_0^t - \int_0^t{\sigma_(\tau) \cdot \frac{dJ(t - \tau)}{d\tau}d \tau}\)</center>

\(= \sigma(t) \cdot J(0) + \int_0^t{\sigma(\tau) \cdot \frac{dJ(t - \tau)}{d(t - \tau)}d\tau}\)


\(\bullet\) Materials response due to temporal variable strains (strain controlled)


<center>\(\sigma(t) = \varepsilon_0 \cdot R(t) + \int_0^t{R(t-\tau) \cdot \frac{d\varepsilon}{d \tau} d\tau}\)</center>

\(= \varepsilon(t) \cdot R(0) + \int_0^t{\varepsilon(\tau) \cdot \frac{dR(t - \tau)}{d(t - \tau)} \cdot d\tau}\)

\(= \int_{-\inf}^{\inf}{R(t - \tau) d\varepsilon}\)

</div>

Informationen

1. Barrierefreie Word Dokumente und einfache PDfs