Material properties

"Optional title"
mid, $\rho$, $E$, $\nu$
$E_{t1}$, $E_{t2}$, $E_{tm}$, $\varepsilon_{tf}$, $\sigma_{ty}$

Parameter definition

Unique material identification number
Young's modulus
Poisson's ratio
Tensile stiffness (lienar term)
Tensile stiffness (quadratic term)
default: not used
Tensile stiffness (maximum)
default: not used
Tensile fiber failure strain
default: no failure
Tensile fiber yield stress
default: no plasticty


This material model is used to model ropes or steel wires.

The stress is defined as:

$\displaystyle{ \boldsymbol{\sigma} = K \mathrm{tr}(\boldsymbol{\varepsilon}) \mathbf{I} + 2G \boldsymbol{\varepsilon}_d + \sum_{i=1}^3 \mathrm{min} \left( \sigma_{ty}, \sigma(\varepsilon_i) \right) \mathbf{v}_i \otimes \mathbf{v}_i }$

where $\boldsymbol{\varepsilon}$ is the total strain, $\boldsymbol{\varepsilon}_d$ is the deviatoric strain tensor, $\varepsilon_i$ is a principal strain and $\mathbf{v}_i$ is its corresponding eigenvector. Note that $K=E/(3(1-2\nu))$ and $G=E/(2(1+\nu))$.

The tensile (fiber) stress $\sigma(\varepsilon_i)$ is defined as:

$\displaystyle{ \sigma(\varepsilon_i) = \left\{ \begin{array}{lcl} 0 & : & \varepsilon_i \leq 0 \\ E_{t1} \varepsilon_i + E_{t2} \varepsilon_i^2 & : & \varepsilon_i \leq \varepsilon_{tm} \\ E_{t1} \varepsilon_{tm} + E_{t2} \varepsilon_{tm}^2 + E_{tm} (\varepsilon_i - \varepsilon_{tm}) & : & \varepsilon_i \gt \varepsilon_{tm} \end{array} \right. }$

$\varepsilon_{tm}$ is the strain where the full tensile stiffness $E_{tm}$ has been reached:

$\displaystyle{ \varepsilon_{tm} = \frac{E_{tm} - E_{t1}}{2E_{t2}} }$

$\varepsilon_{tf}$ is an optional tensile failure strain and $\sigma_{ty}$ is an optional tensile yield stress.