Nonlinear Material Model Damage

Technical Article

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One of my earlier articles described the Isotropic Nonlinear Elastic material model. However, many materials do not have a purely symmetrical nonlinear material behaviour. In this regard, the yield laws according to von Mises, Drucker-Prager and Mohr-Coulomb mentioned in this previous article are also limited to the yield surface in the principal stress space.

Figure 01 - Yield Surfaces in RFEM (von Mises, Tresca, Drucker-Prager, Mohr-Coulomb)

Therefore, these yield rules only allow for the modeling of purely elastic-plastic material behavior. For materials that are subject to a damage process due to cracks, for example, the Material Model Damage described below is more suitable. A good example of such a material is concrete, which has a much higher compressive strength than tensile strength. Cracks that occur in the tension area of the material reduce the stiffness of the system. In the case of reinforced concrete or steel fiber concrete, the reinforcement absorbs the tensile stresses in this case.

Theoretical Background

Nonlinear material models are generally represented by the displacement of a body in the deformed current space to a stress-free reference configuration (see Figure 02). You can find more information about this topic in [2] , for example.

Figure 02 - Kinematic Relation Between Reference and Current Configuration (Source: [1])

A deformation tensor is used to model the deformations of the local element in the reference system. The strain in the undeformed reference system is determined by the Green-Lagrange strain tensor E = ½ ∙ (F T ∙ F - 1) and the strain in the local coordinate system by the Euler-Almansi strain tensor e = ½ ∙ (I - b -1 ) is derived. From these two strains, partial integration is used to derive a linear strain ε = ½ ∙ (H + H T ), from which the nominal stresses on the body can be calculated by means of the Cauchy theorem and the Piola-Kirchhoff stress tensor. Thus, free energy rates can be represented by means of the balance equations of the continuum.

Balance equations of the continuum:

  • The mass balance states that the mass of the system remains the same even if it is deformed.
    $$\mathrm m\;=\;\int_\mathrm{Bt}\mathrm{ρdν}\;(\mathrm{verformtes}\;\mathrm{System})\;=\;\int_{\mathrm B0}\mathrm{dV}\;(\mathrm{Referenzsystem})\;=\;\mathrm{konstant}$$
  • Pulse balance as temporal change of the total momentum
    $$\frac{\mathrm d}{\mathrm{dt}}\int_\mathrm{Bt}\mathrm{ρẋdν}\;=\;\int_\mathrm{Bt}\mathrm{ρbdν}\;(\mathrm{Volumenkräfte})\;+\;\int_\mathrm{δBt}\mathrm{tda}\;(\mathrm{Oberflächenkräfte})$$
  • Angular momentum balance as rate of change of the total momentum
    $$\frac{\mathrm d}{\mathrm{dt}}\;\int_\mathrm{Bt}\mathrm{ρx}\;\cdot\;\mathrm{ẋdν}\;=\;\int_\mathrm{Bt}\mathrm\rho\;(\mathrm x\;\cdot\;\mathrm b)\;\mathrm{dν}\;+\;\int_\mathrm{δBt}\mathrm x\;\cdot\;\mathrm{td}$$
  • 1. Law of Thermodynamics: 1. total energy of a body is retained.
    $$\frac{\mathrm d}{\mathrm{dt}}\int_\mathrm{Bt}(\mathrm u\;+\;\frac12\;\cdot\;\mathrm ẋ\;\cdot\;\mathrm ẋ)\;\mathrm{ρdν}\;=\;\int_\mathrm{Bt}\mathrm{ρr}\;+\;\mathrm b\;\cdot\;\mathrm{ẋdν}\;+\;\int_\mathrm{δBt}\mathrm t\;\cdot\;\mathrm ẋ\;-\;\mathrm q\;\cdot\;\mathrm{nda}$$
    kinetic energy = mechanical power + stress surface
  • Second Law of Thermodynamics: 2.in the transition to another plane, energy (heat) is released.
    $$\frac{\mathrm d}{\mathrm{dt}}\int_\mathrm{Bt}\mathrm{ρsdν}\;\geq\;\int_\mathrm{Bt}\mathrm\rho\;\frac{\mathrm r}{\mathrm\theta}\;\mathrm\nu\;-\;\int_\mathrm{δBt}\frac1{\mathrm\theta}\;\mathrm q\;\cdot\;\mathrm{nda}$$

With the state equations (constitutive equations), the material dependence of bodies is introduced. The damage is considered in the material model by means of the internal variables (free energy ψ, specific entropy s, Cauchy stress tensor σ, heat flow vector q). In this context, the material's "memory" also plays an important role, that is its time-dependent behavior. This is taken into account by means of kinematic and isotropic hardening. With regard to the damage of the material, the distortion component is decomposed additively into an elastic and plastic component. The plastic component from this is again divided into a kinematic and isotropic component.
ε = ε e + ε p → ε p = ε iso + ε kin

In the article about the nonlinear elastic material behavior, it has already been explained that the yielding function, which considers the damage effects, depends on the invariants of the stress tensor. Specifically, the yielding function is subject to a restriction according to the so-called Kuhn-Tucker condition, which states that all stress states within the principal stress range are less than 0 and thus elastic. Stresses outside this space are not allowed and are projected back to the yielding surface by means of a correction step (predictor-corrector step). This calculation is performed by means of a test function, which requires a nonlinear calculation method according to Newton-Raphson.

Figure 03 - Graphical Display of Yield Surface in Principal Stress Space

The yielding function (from [4] ) in the material model Damage distinguishes between tension and compression for the material:
$$\begin{array}{l}\mathrm d^+\;=\;\mathrm g^+\;=\;1\;-\;\frac{\mathrm r_0^+}{\mathrm r^+}\;\cdot\;\left\{(1\;-\;\mathrm A^+)\;+\;\mathrm A^+\;\exp\;\left[\mathrm B^+\;\cdot\;\left(1\;-\;\frac{\mathrm r^+}{\mathrm r_0^+}\right)\right]\right\}\;(\mathrm{Gleichung}\;54,\;\mathrm{Zug})\\\mathrm d^-\;=\;\mathrm g^-\;=\;1\;-\;\frac{\mathrm r_0^-}{\mathrm r^-}\;\cdot\;\left\{(1\;-\;\mathrm A^-)\;+\;\mathrm A^-\;\exp\;\left[\mathrm B^-\;\cdot\;\left(1\;-\;\frac{\mathrm r^-}{\mathrm r_0^-}\right)\right]\right\}\;(\mathrm{Gleichung}\;58,\;\mathrm{Druck})\\\rightarrow\;\mathrm d^{+/-}\;=\;\mathrm r^{+/-}\;\cdot\;\mathrm h^{+/-}\;\leq\;0\end{array}$$

Use r to designate the energy rate and h to harden the function. The variables A and B describe the damage of the material. As in the next chapter, this is also done via the stress-strain relation in the principal stress area.

Damage in RFEM

After this basic introduction to the topic, the handling of the material model in RFEM will be explained. Within the scope of this article, only a rough overview of the topic is possible, which may also have gaps in connection. Therefore, further literature such as [2] is recommended.

Due to the nonlinear calculation method with the correction step, the calculation in the first step of the diagram must be linearly elastic. The solution in RFEM provides that the strain in the second step of the diagram depends on the modulus of elasticity defined in the Material dialog box and the defined limit stress (see Figure 04).

Figure 04 - Entering Stress-Strain Diagram in RFEM

The strain is subject to Hooke's law ε = σ/E. After this first elastic predictor step, it is possible to define almost any antimetric definition of the stress-strain diagram. In this case, it is also possible that the modulus of elasticity of the material becomes negative because it is calculated back using the following relation:
$$\frac{{\mathrm\sigma}_\mathrm i\;-\;{\mathrm\sigma}_{\mathrm i-1}}{{\mathrm\varepsilon}_\mathrm i\;-\;{\mathrm\varepsilon}_{\mathrm i-1}}\;=\;\mathrm E$$

However, since the modulus of elasticity is only necessary for the retroactive calculation of the relation, the module's value is also possible here. The calculation by means of a correction iteration, which is also described above, ensures that the stiffness of the system is reduced in the material model Damage until each individual FE element no longer absorbs any stress. The strains in the respective element can become very large.

Summary

With the material model Damage, it is possible to perform the nonlinear calculation with antimetric, almost arbitrary stress-strain relations. If the material is damaged, however, the system remains a continuum. This means that no cracks are displayed in the system. The numerical effort for this would be very considerable. It would be necessary, for example, to re-mesh the system with a so-called adaptive FE mesh. Due to the restrictions mentioned above, very large strains can build up in the system.

In case of very high distortion, the user should perform a division of the system manually. This could be done, for example, by using contact solids with corresponding yield strengths. Furthermore, a plastic distortion of the element is not considered by this material model. This can be especially helpful in the print area. For the usual problem of cracked concrete in the tension area, the material model is sufficiently accurate.

Reference

[1]   Barth, C .; Rustler, W .: Finite Elements in Structural Engineering, 2nd Edition. Berlin: Beuth, 2013
[2]  Nackenhorst, U .: Lecture notes Solid Mechanics. Hanover: Institute of Mechanics and Numerical Mechanics, Gottfried Wilhelm Leibniz University, 2015
[3]  Altenbach, H .: Continuum Mechanics - Introduction to Material-Independent and Material-Dependent Equations, 3rd Edition. Berlin: Springer, 2015
[4] Hürkamp, A .: Micro-Mechanically Based Damage Analysis of Ultra High Performance Fiber Reinforced Concrete Structures with Uncertainties. Hanover: Institute of Mechanics and Numerical Mechanics, Gottfried Wilhelm Leibniz University, 2013
[5] Wu, J. Y .; Li J .; Faria R. An energy release rate-based plastic-damage model for concrete, International Journal of Solids and Structures 43, pages 583 - 612. Amsterdam: Elsevier, 2006

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