General and Specific Quadratic Yield Functions

Comparative evaluation of non-associated quadratic and associated quartic plasticity models for orthotropic sheet metals

A plastic constitutive model for initially orthotropic but subsequently skewed anisotropic materials is proposed in this paper. The variation of anisotropic axes is discussed within the framework of objectivity, and the evolution of the axes is formulated. It is shown that the yield function proposed is identical to Hill's quadratic yield function when the anisotropic axes lie along the base vectors. Applying the conventional laws for both work-hardening and material flow to the yield function, an anisotropic constitutive model is completed. A finite element equation is obtained next, based on the updated Lagrangian formulation. A fully three-dimensional analysis is carried out to predict the strain localization behavior of anisotropic sheet metals. The direction of strain localization is strongly affected by the initial configuration of the anisotropic axes, and the simulated result shows fairly good agreement with the experimental one.

The aim of this work is to determine what free energy functionals are expressible as quadratic forms of the state functional \(I^t\) which is discussed in earlier papers. The single integral form is shown to include the functional \(\psi _F\) proposed a few years ago, and also a further category of functionals which are easily described but more complicated to construct. These latter examples exist only for certain types of materials. The double integral case is examined in detail, against the background of a new systematic approach developed recently for double integral quadratic forms in terms of strain history, which was used to uncover new free energy functionals. However, while, in principle, the same method should apply to free energies which can be given by quadratic forms in terms of \(I^t\), it emerges that this requirement is very restrictive; indeed, only the minimum free energy can be expressed in such a manner.

In this work the onset of failure caused by particle-matrix debonding in aluminum alloy 2090-T3 and magnesium alloy AZ31 under either overall uniaxial plane strain tension or compression load is studied numerically using a unit cell model. Especially for magnesium AZ31, experiments show a significant difference in the tensile-compressive yield stresses, the so-called Strength-Differential-effect (SD-effect). This SD-effect cannot be captured by any isotropic yield function and only by recent anisotropic yield functions. To accurately model AA 2090-T3 and magnesium AZ31 the general yield function by Yoon et al. (2014) is adopted in addition to the isotropic von Mises plasticity yield function and the anisotropic yield function of Hill (1948). The numerically predicted strain at which void nucleation initiates is generally found to be larger for magnesium AZ31 than for AA 2090-T3. However, depending on the orientation of the principal axes of plastic anisotropy this void nucleation strain varies significantly. This holds true for both materials, but the void nucleation strain of AA 2090-T3 in compression is numerically predicted to be practically identical whether or not isotropic or anisotropic plasticity is considered. In tension the void nucleation strain of AA 2090-T3 based on anisotropic plasticity is found to be more than double the strain obtained by von Mises plasticity. The numerically predicted void nucleation strain of magnesium AZ31 is significantly affected by the yield function adopted. For the von Mises yield function the value of the void nucleation strain is in between the predictions based on the anisotropic yield functions. For both tension and compression, the yield function of Yoon et al. (2014) results in the largest void nucleation strain, whereas the yield function of Hill results in the smallest strain value. This holds true for all five orientations of the principal axes of plastic anisotropy investigated.

Creasing and offset printing are both examples of paperboard converting operations where the stress state is multiaxial, and where elastic-plastic deformation occurs in the thickness direction. Optimization of paperboard for such operations requires both advanced modeling and a better understanding of the mechanical behavior of the material. Today, our understanding and modeling of the out-of-plane properties are not as well established as our knowledge of the in-plane behavior. In order to bridge this gap, a modification of the Arcan device, which is well known in other fields, was developed for the experimental characterization of the out-of-plane mechanical behavior of paperboard. A fixture attached to the Arcan device was used to control the deformation in the test piece during loading. The test piece was glued to the device with a high viscosity adhesive and left stress-free during curing to achieve an initial state free of stresses. The apparatus proved to work well and to produce reliable results. Measurements of the mechanical behavior in combined normal and shear loading generated data points for the determination of the yield surface in the stress space. The elastic-plastic behavior in the thickness direction of paperboard was modeled assuming small-strain orthotropic linear elasticity and a quadratic yield function. Simulations using this yield function and an associative flow law showed good agreement with the test results.

An accurate description of the directional dependence of uniaxial tensile yielding and plastic flow in advanced high-strength steel sheets may require either a nonassociated plasticity model with separate quadratic yield function and flow potential or an associated plasticity model with nonquadratic yield function. In this paper, Gotoh's fourth-order homogeneous polynomial yield function is applied to model two advanced high-strength steel sheets in an associated plasticity model. Both the parameter selection for calibrating Gotoh's yield function and its positivity and convexity verification are given in some detail. Similarities and differences between the associated plasticity model presented here and the nonassociated one appeared in the literature are discussed in terms of the directional dependence of yield stresses and plastic strain ratios under uniaxial tension and yield stresses under biaxial tension loading.

Energy-stable boundary conditions based on a quadratic form: Applications to outflow/open-boundary problems in incompressible flows

Design of linkage mechanisms of partially rigid frames using limit analysis with quadratic yield functions

Quadratic functions on torsion groups

Results of calculations to determine specific yield functions for IGY neutron monitors are reported for vertically incident cosmic-ray protons in the range of 1 < Pc < 6 GV. The Monte Carlo method is used to calculate secondary nucleon-flux intensities at atmospheric depths of 312, 680, and 1 000 g/cm2. Experimental data on neutron monitor response are employed to relate the calculated flux data to count rates, and hence to specific yields, at the corresponding depths. Comparison with empirical data for the specific yield function at sea level shows fair agreement for rigidities less than 3 GV, but calculated yields are considerably smaller than the empirical data at higher rigidities. Various possible explanations for the discrepancy are discussed. At this time, the most probable cause appears to be the assumption that vertically incident cosmic rays produce most of the counts registered by neutron monitors.

A peridynamic approach to computation of elastic and entropic interactions of inclusions on a lipid membrane

The analysis of linear elastic structures having cable-like members, that is members which can take tension but not compression, is formulated as one in quadratic programming. The objective function is the total complementary energy function, a quadratic form in terms of the variables which are forces and force resultants, and the constraints are the equations of equilibrium of forces at the joints. A technique is indicated to reduce the size of the quadratic program obtained so that the number of variables in the reduced program is approximately the larger of the degree of indeterminacy and the number of the cable-like members. The formulation is performed entirely for a truss.

We consider two quadratic forms on a class of manifolds conformally equivalent to the euclidean space. We give necessary and sufficient conditions for the equivalence of these quadratic forms in terms of a weighted Hardy inequality.

Minkowski established a lower bound for the determinant D of a Minkowski-reduced quadratic form in terms of the product of its diagonal coefficients aii (i = 1, …n). Oppenheim and Barnes found, for n = 3 and n = 4 respectively, the precise minimum of D in terms of the aii; in each case the minimum is a polynomial in the aii. Here it is shown that no such result exists when n = 5; however a polynomial in all,…, a55 is determined which gives the minimum of D when a55 is sufficiently large.

A nonlinear programming approach to kinematic shakedown analysis of frictional materials