Mathias Wallin

Strain mapping in free-standing heterostructured wurtzite InAs/InP nanowires

The strain distribution in heterostructured wurtzite InAs/InP nanowires is measured by a peak finding technique using high resolution transmission electron microscopy images. We find that nanowires with a diameter of about 20 nm show a 10 nm strained area over the InAs/InP interface and the rest of the wire has a relaxed lattice structure. The lattice parameters and elastic properties for the wurtzite structure of InAs and InP are calculated and a nanowire interface is simulated using finite element calculations. Both the method and the experimental results are validated using a combination of finite element calculations and image simulations.

Kinematic hardening in large strain plasticity

A finite strain hyper elasto-plastic constitutive model capable to describe non-linear kinematic hardening as well as nonlinear isotropic hardening is presented. In addition to the intermediate configuration and in order to model kinematic hardening, an additional configuration is introduced - the center configuration; both configurations are chosen to be isoclinic. The yield condition is formulated in terms of the Mandel stress and a back-stress with a structure similar to the Mandel stress. It is shown that the non-dissipative part of the plastic velocity gradient not governed by the thermodynamical framework and the corresponding quantity associated with the kinematic hardening influence the material behaviour to a large extent when kinematic hardening is present. However, for isotropic elasticity and isotropic hardening plasticity it is shown that the non-dissipative quantities have no influence upon the stress-strain relation. As an example, kinematic hardening von Mises plasticity is considered, which fulfils the plastic incompressibility condition and is independent of the hydrostatic pressure. To evaluate the response and to examine the influence of the non-dissipative quantities, simple shear is considered; no stress oscillations occur. (Less)

Modeling of the Degradation of Elastic Properties due to the Evolution of Ductile Damage

An elasto-plastic constitutive model for porous materials is formulated within the thermodynamic framework. The formulation facilitates a natural modeling of damage as well as growth and the shrinkage of voids. Metal plasticity is used for demonstrating the possibilities of the formulation. The yield function employed is assumed to depend upon the void-volume fraction, whereas the free energy is dependent on a scalar damage field. To show the capabilities of the model the algorithmic constitutive equations are derived and implemented into a finite element program. It is shown that an extremely simple system involving only two scalar equations needs to be solved in the constitutive driver. Two numerical examples are considered: the necking of an axi-symmetric bar and localization in a notched specimen.

Numerical integration of elasto-plasticity coupled to damage using a diagonal implicit Runge–Kutta integration scheme

This article is concerned with the numerical integration of finite strain continuum damage models. The numerical sensitivity of two damage evolution laws and two numerical integration schemes are investigated. The two damage models differ in that one of the models includes a threshold such that the damage evolution is suppressed until a certain effective plastic strain is reached. The classical integration scheme based on the implicit Euler scheme is found to suffer from a severe step-length dependence. An alternative integration scheme based on a diagonal implicit Runge--Kutta scheme originally proposed by Ellsiepen (1999) is investigated. The diagonal implicit Runge--Kutta scheme is applied to the balance of momentum as well as the constitutive evolution equations. When applied to finite strain multiplicative plasticity, the diagonal implicit Runge--Kutta scheme destroys the plastic incompressibility of the underlying continuum evolution laws. Here, the evolution laws are modified such that the incompressibility of the plastic deformation is preserved approximately. The presented numerical examples reveal that a significant increase in accuracy can be obtained at virtually no cost using the diagonal implicit Runge--Kutta scheme. It is also shown that for the model including a discontinuous evolution law, the superiority of the diagonal implicit Runge--Kutta scheme over the implicit Euler scheme is reduced.

Modeling of the long-term behavior of glassy polymers

The constitutive model for glassy polymers proposed by Arruda and Boyce (BPA model) is reviewed and compared to experimental data for long-term loading. The BPA model has previously been shown to capture monotonic loading accurately, but for unloading and long-term behavior, the response of the BPA model is found to deviate from experimental data. In the present paper, we suggest an efficient extension that significantly improves the predictive capability of the BPA model during unloading and long-term recovery. The new, extended BPA model (EBPA model) is calibrated to experimental data of polycarbonate (PC) in various loading-unloading situations and deformation states. The numerical treatment of the BPA model associated with the finite element analysis is also discussed. As a consequence of the anisotropic hardening, the plastic spin enters the model. In order to handle the plastic spin in a finite element formulation, an algorithmic plastic spin is introduced. In conjunction with the backward Euler integration scheme use of the algorithmic plastic spin leads to a set of algebraic equations that provides the updated state. Numerical examples reveal that the proposed numerical algorithm is robust and well suited for finite element simulations. (Less)

The influence of non-dissipative quantities in kinematic hardening plasticity

A kinematic hardening plasticity model valid for finite strains is presented. The model is based on the well-known multiplicative split of the deformation gradient into elastic and plastic parts. The basic ingredient in the formulation is the introduction of a locally defined configuration—a centre configuration—which is associated with a deformation gradient that is used to characterize the kinematic hardening behaviour. The non-dissipative quantities allowed in the model are found when the plastic and kinematic hardening evolution laws are split into two parts: a dissipative part, which is restricted by the dissipation inequality, and a non-dissipative part, which can be chosen without any thermodynamic considerations. To investigate the predictive capabilities of the proposed kinematic hardening formulation, necking of a bar is considered. Moreover, to show the influence of the non-dissipative quantities, the simple shear problem and torsion of a thin-walled cylinder are considered. The numerical examples reveal that the non-dissipative quantities can affect the response to a large extent and are consequently valuable and important ingredients in the formulation when representing real material behaviour.

A Constitutive Model for Ductile Damage Evolution

A constitutive model for ductile porous material is formulated within the thermodynamic framework. A yield function based on the lower-bound solution for a cylindrical void model embedded in a plastic matrix is proposed. The new yield function is compared to the classical Gurson yield function using cell model calculations. The results reveal that the proposed yield function agreed well with the plastic region found from the cell model calculations. In addition to the influence of the void-volume ratio, the elastic part of the free energy is dependent on a scalar damage field which allows the elasticity to be influenced by the void-volume fraction. The degradation is controlled by a scalar valued damage field and enters the formulation via the Helmholtzs free energy. This dependence allows the elastic properties to naturally depend upon the damage accumulation. The numerical treatment of the model is derived and the capability of the model is demonstrated via numerical simulation of the necking of an axi-symmetric bar.

The Influence of Non-Dissipative Quantities in Kinematic Hardening Plasticity

A kinematic hardening plasticity model valid for finite strains is presented. The model is based on the well-known multiplicative split of the deformation gradient into an elastic and a plastic part. The basic ingredient in the formulation is the introduction of locally defined configurations center configurations- which are associated with deformation gradients that are used to characterize the kinematic hardening behavior. One of the aspects of the model investigated here is found when the plastic and kinematic hardening evolution laws are split into two parts: a dissipative part, which is restricted by the dissipation inequality, and a non-dissipative part, which can be chosen without any thermodynamical considerations. To investigate the predictive capabilities of the proposed formulation, the simple shear problem and torsion of a thin-walled cylinder are considered. In the numerical examples it turns out that the non-dissipative quantities affect the response to a large extent and are consequently valuable ingredients in the formulation when representing real material behavior (Less)