Analysis of sheet metal forming operations by a stress resultant constitutive law

Sheet metal forming is simulated by finite element methods using a stress resultant constitutive law in this paper. A Lagrangian description of axisymmetric and plane-strain shell deformation is first reviewed. Then a stress resultant constitutive law in rate form is presented, where the effect of thickness reduction due to large plastic deformation is considered. A finite element formulation in terms of stress resultants and their work-conjugate generalized strain rates is derived based on the virtual work principle. A hemispherical punch stretching operation and a plane-strain draw operation are simulated by a finite element program based on the finite element formulation. The results of these finite element simulations are in good-agreement with those using the through-the-thickness integration method. The results of the hemispherical punch stretching simulation suggest that the coupling term of moments and membrane forces of the modified Ilyushin yield function should be eliminated to avoid numerical instability under stretching-dominated conditions for this rate-independent plasticity formulation. Further, the results suggest that the hardening rule in a power-law form based on the small-strain approach must be modified to take account for finite deformation effects of combined stretching and bending. Under the plane-strain draw operation, the sheet experiences a large amount of bending before the final stretching. The simulation based on the stress resultant constitutive law can produce this essential aspect of deformation pattern as that of the through-the-thickness integration method, whereas a simulation based on a membrane theory cannot. In conclusion, the results of these simulations indicate that a finite element program based on the stress resultant constitutive law can simulate sheet-forming processes with much shorter computational time than that based on the through-the-thickness integration method.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/50100/1/1620370502_ftp.pd

Plane-strain crack-tip fields for power-law hardening orthotropic materials

Near-tip stress and strain fields for power-law hardening orthotropic materials under plane-strain conditions are presented. Plastic orthotropy is described by Hill's quadratic yield function. The angular variations of these HRR-type fields depend on a single parameter which specifies the state of plastic orthotropy. Near-tip fields for highly orthotropic materials differ substantially from the fields for isotropic materials. Mode I (symmetric) and mode II (anti-symmetric) solutions for different degrees of plastic orthotropy are given. The angular stress distributions for the low-hardening material agree remarkably well with the plane-strain slip-line fields. Based on the singularity fields, effective stress contours are constructed. The applicability of these fields in the context of a fiber-reinforced composite containing a macroscopic flaw is discussed.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/25953/1/0000019.pd

Plane-stress crack-tip fields for power-law hardening orthotropic materials

Plane stress mode I near-tip fields in orthotropic materials are examined. Plastic orthotropy is described by Hill's quadratic yield function and the strain hardening behavior is given by an appropriate generalization of a uniaxial tensile power-law stress-strain relation. Pronounced changes in the pattern of the angular variations of crack-tip fields have been observed with the degree of plastic orthotropy and the amount of strain hardening. Possible shapes and sizes of plastic zones (as inferred from effective stress contours) are presented for high- and low-hardening materials and a wide range of plastic orthotropy. The shape of the plastic zone for a particular case of plastic orthotropy agreed remarkably well with the zone of intense straining induced by an appropriately orientated crack within a graphite/epoxy laminate. On examine les champs de contraintes planes selon un mode I au voisinage de l'extrémité d'une fissure dans des matériaux orthotropes. L'orthotrope plastique est décrite par la fonction quadratique de plastification de Hill, et le comportement à l'écrouissage est donné par une généralisation adéquate d'une relation tensioncilatation de forme parabolique, sous traction mono-axiale. On a observé des modifications profondes dans l'aspect des variations angulaires des champs d'extrémité de fissure, selon le degré d'orthotropie plastique et infensité de l'écrouissage. Pour des matériaux très sujets ou peu sujets à l'écrouissage, et pour une large gamme d'orthotropies plastiques, on présente les formes et dimensions possibles des zones plastiques, telles qu'elles se deduisent des contours effectifs de contraintes. La forme de la zone plastique correspondant au cas particulier d'une orthotropie plastique s'accorde remarquablement bien à la zone de dilatation importante créée par une fissure d'orientation appropriée, dans une plaque de graphite-epoxy.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42768/1/10704_2004_Article_BF00045862.pd

A fracture parameter for welded structures with residual stresses

In this paper, a finite element procedure to determine a fracture parameter J ld is presented for welded structures with consideration of residual stresses. The method is based on the energy difference of two cracked solids with slightly different crack sizes. Our computational results show that J ld and the J integral agree well for a cracked plate without consideration of residual stresses. When the residual stresses are considered, the values of J ld for different contours close to the crack tip in the cracked plate subject to remote tensile stresses are in good agreement. The computational results also indicate that for the given residual stress distribution, the values of J ld with consideration of residual stresses are lower than those without consideration of residual stresses for the cracked plate subject to large remote tensile stresses.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42294/1/466-22-3-281_80220281.pd

Elastic-plastic analysis of combined mode I, II and III crack-tip fields under small-scale yielding conditions

Within the context of the small-strain approach, combined mode I, II and III near-tip fields of stationary cracks in power-law hardening materials are investigated. We use a finite element technique to obtain asymptotic angular stress solutions for combined mode I and II perturbed from mode III. These perturbation solutions with the same stress singularities as those of pure mode III are presented for different hardening materials. The perturbation results further suggest that the order of crack-tip stress singularities varies smoothly with changing mode mixity. We also employ full-field finite element compulations to study the small-scale yielding near-tip fields for several combinations of prescribed remote mode I. II and III elastic K fields. These solutions verify an interesting pattern which agrees with the previous solutions for combined mode I and III loading as well as those for combined mode II and III loading: well within the plastic zone, under near mode I mixed-mode loadings, the in-plane stresses are slightly more singular than r-t/(n + 1) while the out-of-plane shear stresses are slightly less singular than r-t/(n + 1), where r is the radial distance to the lip and n is the strain hardening exponent of the material. To explain the complex behavior of the near-tip stresses, we introduce an effective in-plane shear stress and an effective out-of-plane shear stress to quantify the in-plane and out-of-plane plastic shear at different orientations in a consistent manner. The full-field solutions also corroborate the observation that the singularities of the in-plane slresses and the out-of-plane shear stresses vary smoothly with mode mixity.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/30377/1/0000779.pd

Elastic-plastic analysis of combined mode I and III crack-tip fields under small-scale yielding conditions

W context of the small-strain approach, combined mode I and III near-tip fields of a stationary crack in an elastic-plastic solid are obtained by finite element analysis under small-scale yielding conditions. To investigate the behavior of the near-tip fields, the normalized stresses ahead of the crack tip are plotted as functions of the normalized radial distance to the tip for several combinations of prescribed mode I and III elastic K fields. The angular variations of the normalized stresses at a fixed radial distance deep within the plastic zone are also plotted for several combinations of remote mode I and III elastic K fields. These plots show an unmistakeable pattern : the in-plane stresses are more singular than the out-of-plane shear stresses. Over a certain distance, the near-tip in-plane stresses can be said to be more singular than r-1/(n + 1) while the near-tip out-of-plane shear stresses are less singular than r-1/(n + 1), where r is the radial distance to the tip and n is the strain hardening exponent of the material. Implications of these features as they relate to three-dimensional engineering fracture analyses are discussed.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/28826/1/0000660.pd

Constitutive laws for thin plates of power-law materials

With the Kirchhoff assumption, a stress resultant constitutive law as a function of the kinematic variables for thin plates of power-law hardening materials is derived under proportional straining conditions. Also, in analogy to the Jz deformation plasticity and incremental plasticity theory, a flow rule, based on the constitutive law, to describe the elastic plastic behavior of the plate is proposed. The constitutive behavior of the plates subjected to uniaxial combined membrane force and bending moment is examined in detail and the results are compared with those for the corresponding clastic plastic materials using the through-the-thickness integration method. The yield surfaces for power-law materials are constructed and the formation of vertiees on the yield surfaces for perfectly plastic materials under such a loading condition is clearly demonstrated.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/29666/1/0000755.pd

Nonproportional loading effects on elastic-plastic behavior based on stress resultants for thin plates of strain hardening materials

A stress resultant constitutive law in rate form is constructed for power-law hardening materials. The change of plate thickness is considered in the constitutive law. The elastic-plastic behavior of a plate element based on the stress resultant constitutive law under uniaxial combined tension and bending is determined under a limited number of nonproportional and unloading paths. The results based on the stress resultant constitutive law and the through-the-thickness integration method are compared within the context of both the small-strain and finite deformation approaches. The results indicate that the selection of the normalized equivalent stress resultant and the corresponding work-conjugate normalized equivalent generalized strain is appropriate for describing the hardening behavior in the stress resultant space. However, the hardening rule in a power law form must be modified for low hardening materials at large plastic deformation when finite deformation effects are considered.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/31919/1/0000872.pd