High-order time integration applied to metal powder plasticity

Hartmann, Stefan; Bier, Wolfgang

doi:10.1016/j.ijplas.2007.01.014

Cited by 37 publications

(21 citation statements)

References 58 publications

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“…A relatively new single‐surface yield criterion applicable to metal powder compaction as well as granular materials and foams is the model proposed by Bier and Hartmann . The yield function consists of a so‐called log‐log‐interpolation of 2 simple convex yield functions, where an ellipse yield function is combined with an exponential function resulting in a single‐face yield criterion:

\begin{matrix} alignleft \\ align-1 F (I_{1}, \sqrt{J_{2}}) & align-2 = \sqrt{J_{2}} + c \ln (\frac{\exp (- f_{1} (I_{1}) / c) + \exp (- f_{2} (I_{1}) / c)}{2}) . \end{matrix}

The ellipsoidal part f 1 ( I 1 ) and the exponential part f 2 ( I 1 ) of the yield function

F false(I_{1}, \sqrt{J_{2}} false)

are defined as follows:

\begin{matrix} alignleft \\ align-1 f_{1} (I_{1}) & align-2 = \sqrt{k^{2} - α {(I_{1} - 3 ξ)}^{2}} with k = \sqrt{α {(I_{0} - 3 ξ)}^{2}}, \end{matrix}

\begin{matrix} alignleft \\ align-1 f_{2} (I_{1}) & align-2 = A_{1} - A_{2} \exp (A_{3} I_{1...} \end{matrix}

…”

Section: Yielding Behavior Of Cellular Materialsmentioning

confidence: 99%

Yield surfaces for solid foams: A review on experimental characterization and modeling

Jung

Diebels

2018

GAMM-Mitteilungen

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Solid foams are a very important class of lightweight materials with energy absorption properties for application in automotive and aerospace industry. In such applications, foams are subjected not only to uniaxial but in most cases to very complex multiaxial stress states. Hence, yield surfaces are needed for the reliable prediction of the failure behavior by simulations under multiaxial loading. As multiaxial loading of cellular solids is a very challenging task, in literature, yield surfaces are usually based mainly on simulation results not validated by real experimental data sets. The present contribution starts with a review of the theory of yield surfaces and numerical models. It gives a deep insight into the state of the art on the experimental assessment of yield surfaces for solid foams. Finally, a yield surface and its post‐yield surfaces are determined exemplarily for open‐cell aluminum foams by using a multitude of experimental methods.

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\begin{matrix} alignleft \\ align-1 F (I_{1}, \sqrt{J_{2}}) & align-2 = \sqrt{J_{2}} + c \ln (\frac{\exp (- f_{1} (I_{1}) / c) + \exp (- f_{2} (I_{1}) / c)}{2}) . \end{matrix}

The ellipsoidal part f 1 ( I 1 ) and the exponential part f 2 ( I 1 ) of the yield function

F false(I_{1}, \sqrt{J_{2}} false)

are defined as follows:

\begin{matrix} alignleft \\ align-1 f_{1} (I_{1}) & align-2 = \sqrt{k^{2} - α {(I_{1} - 3 ξ)}^{2}} with k = \sqrt{α {(I_{0} - 3 ξ)}^{2}}, \end{matrix}

\begin{matrix} alignleft \\ align-1 f_{2} (I_{1}) & align-2 = A_{1} - A_{2} \exp (A_{3} I_{1...} \end{matrix}

…”

Section: Yielding Behavior Of Cellular Materialsmentioning

confidence: 99%

Yield surfaces for solid foams: A review on experimental characterization and modeling

Jung

Diebels

2018

GAMM-Mitteilungen

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“…and stored for the calculation (30) and finally for (27). The coefficients of an SDIRK-method are compiled in the Butcher-tableau, see Table 1.…”

Section: +K(t θ)θ(T)−p θ (T θ) = 0 (23)mentioning

confidence: 99%

“…The derivation follows the ideas in [13,[26][27][28], where the methods are applied to purely mechanical problems. This procedure has the advantage that backward Euler based finite element implementations are easily extendable to a high-order DIRK-method.…”

Section: Introductionmentioning

confidence: 99%

Experimental validation of high-order time integration for non-linear heat transfer problems

et al. 2011

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An accurate prediction of the temperature distribution in space and time plays an important role in many industrial applications, in particular when phase transformations are involved. In this article the thermo-physical properties of steel 51CrV4 (SAE 6150) are determined and used in numerical simulations. For the simulation of the temperature field a semi-discrete approach is used, consisting of a finite element approximation in space and a high order RungeKutta integration in time. Several adaptive high-order time integration method (stiffly accurate diagonally implicit Runge-Kutta methods) are applied and their computational efficiency is investigated. The theoretical rates of convergence are achieved for all problems, including the non-linear case. Whereas the second order accurate method of Ellsiepen with time adaptive step-size control proves to be most efficient. Further, the influence of the material model on the simulation results is studied and the numerical results are verified by experiments. The best correlation of the simulation and experimental data is achieved using temperature-dependent parameters.

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“…), see for different applications [28,30,31]. Thereby, u ∈ R nu are the unknown nodal displacements and q ∈ R nQ the internal variables at all spatial integration points.…”

Section: F(t Y(t)ẏ(t)) := G(t U(t) Q(t)) Q(t) − R(t U(t)u(t) Qmentioning

confidence: 99%

“…Further problems might occur in constitutive equations with softening behavior. In this case, local integration steps (8) have to be treated with the methods discussed above, see the discussion in [31] for a pressure sensitive viscoplasticity model. A small viscosity is commonly applied to regularize the case distinctions in yield function based models, see the discussion in [54].…”

Section: F(t Y(t)ẏ(t)) := G(t U(t) Q(t)) Q(t) − R(t U(t)u(t) Qmentioning

confidence: 99%

Iterative solvers within sequences of large linear systems in non‐linear structural mechanics

Hartmann¹,

Tebbens

Quint

et al. 2009

Z Angew Math Mech

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This article treats the computation of discretized constitutive models of evolutionary-type (like models of viscoelasticity, plasticity, and viscoplasticity) with quasi-static finite elements using diagonally implicit Runge-Kutta methods (DIRK) combined with the Multilevel-Newton algorithm (MLNA). The main emphasis is on promoting iterative methods, as opposed to the more traditional direct methods, for solving the non-symmetric systems which occur within the DIRK/MLNA. It is shown that iterative solution of the arising sequences of linear systems can be substantially accelerated by various techniques that aim at sharing part of the computational effort throughout the sequence. In this way iterative solution becomes attractive at clearly lower dimensions than the dimensions where direct solvers start to fail for memory reasons. The applications are related to small strain viscoplasticity of a smooth constitutive model for plastics and a finite strain plasticity model with non-linear kinematic hardening developed for metals.

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High-order time integration applied to metal powder plasticity

Cited by 37 publications

References 58 publications

Yield surfaces for solid foams: A review on experimental characterization and modeling

Yield surfaces for solid foams: A review on experimental characterization and modeling

Experimental validation of high-order time integration for non-linear heat transfer problems

Iterative solvers within sequences of large linear systems in non‐linear structural mechanics

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