Otto T. Bruhns

Logarithmic strain, logarithmic spin and logarithmic rate

SummaryTwo yet undiscovered relations between the Eulerian logarithmic strain inV and two fundamental mechanical quantities, the stretching and the Cauchy stress, are disclosed. A new spin tensor and a new objective tensor-rate are accordingly introduced. Further, new rate-form constitutive models based on this objective tensor-rate are established. It is proved that(i).an objective corotational rate of the logarithmic strain inV can be exactly identical with the stretching and in all strain measures only inV enjoys this property, and(ii).InV and the Cauchy stress σ form a work-conjugate pair of strain and stress. These properties of in ν are shown to determine a unique smooth spin tensor called logarithmic spin and by virtue of this spin a new tensor-rate called logarithmic rate is proposed. In all possible rate-form constitutive models relating the same kind of objective corotational rates of an Eulerian stress measure and an Eulerian strain measure, it is proved that the logarithmic rate is the only choice and the strain measure must be the logarithmic strain inV if the stretching, as is commonly assumed, is used to measure the rate of change of deformation. As an illustration, it is shown that all finite deformation responses of the grade-zero hypoelastic model based on the logarithmic rate are completely in agreement with those of a finite deformation elastic model and moreover this simplest rate-form constitutive model based on the logarithmic rate can predict the phenomenon of the known hypoelastic yield at simple shear.

Self-consistent Eulerian rate type elasto-plasticity models based upon the logarithmic stress rate

Abstract The objective of this article is to suggest new Eulerian rate type constitutive models for isotropic finite deformation elastoplasticity with isotropic hardening, kinematic hardening and combined isotropic-kinematic hardening etc. The main novelty of the suggested models is the use of the newly discovered logarithmic stress rate and the incorporation of a simple, natural explicit integrable-exactly rate type formulation of general hyperelasticity. Each new model is thus subjected to no incompatibility of rate type formulation for elastic behaviour with the notion of elasticity, as encountered by any other existing Eulerian rate type model for elastoplasticity or hypoelasticity. As particular cases, new Prandtl-Reuss equations for elastic-perfect plasticity and elastoplasticity with isotropic hardening, kinematic hardening and combined isotropic-kinematic hardening, respectively, are presented for computational and practical purposes. Of them, the equations for kinematic hardening and combined isotropic–kinematic hardening are, respectively, reduced to three uncoupled equations with respect to the spherical stress component, the shifted stress and the back-stress. The effects of finite rotation on the current strain and stress and hardening behaviour are indicated in a clear and direct manner. As illustrations, finite simple shear responses for the proposed models are studied by means of numerical integration. Further, it is proved that, among all possible (infinitely many) objective Eulerian rate type models, the proposed models are not only the first, but unique, self-consistent models of their kinds, in the sense that the rate type equation used to represent elastic behaviour is exactly integrable to really deliver an elastic relation. ©

Hypo-Elasticity Model Based upon the Logarithmic Stress Rate

Recently these authors have proved [46, 47] that a smooth spin tensor Ωlog can be found such that the stretching tensor D can be exactly written as an objective corotational rate of the Eulerian logarithmic strain measure ln V defined by this spin tensor, and furthermore that in all strain tensor measures only ln V enjoys this favourable property. This spin tensor is called the logarithmic spin and the objective corotational rate of an Eulerian tensor defined by it is called the logarithmic tensor-rate. In this paper, we propose and investigate a hypo-elasticity model based upon the objective corotational rate of the Kirchhoff stress defined by the spin Ωlog, i.e. the logarithmic stress rate. By virtue of the proposed model, we show that the simplest relationship between hypo-elasticity and elasticity can be established, and accordingly that Bernsteins integrability theorem relating hypo-elasticity to elasticity can be substantially simplified. In particular, we show that the simplest form of the proposed model, i.e. the hypo-elasticity model of grade zero, turns out to be integrable to deliver a linear isotropic relation between the Kirchhoff stress and the Eulerian logarithmic strain ln V, and moreover that this simplest model predicts the phenomenon of the known hypo-elastic yield at simple shear deformation.

Bounds to bifurcation stresses in solids with non-associated plastic flow law at finite strain☆

Abstract In the present paper, Hills theory of bifurcation and stability in solids obeying normality is generalized to include a non-associated flow law. A one-parameter family of linear comparison solids has been found that admits a potential and has the property that if uniqueness is certain for the comparison solid then bifurcation and instability are precluded for the underlying elastic-plastic solid. The uniqueness criterion derived may be used as a device to determine lower bounds to the magnitudes of primary bifurcation and instability stresses which are ordinarily unknown. A second linear solid is introduced whose constitutive relations have the same form as the elastic-plastic solid “in loading”. The first eigenstate of this solid gives an upper bound to the primary bifurcation state of the underlying elastic-plastic solid. The search for the genuine primary bifurcation state is therefore replaced by a search for upper and lower bounds in the situation when normality fails to hold. The theory is applied to problems of homogeneous stress states.

On objective corotational rates and their defining spin tensors

Abstract In this paper, we prove a general result on objective corotational rates and their defining spin tensors: let Ω* be a spin tensor that is associated with the rotation and deformation of a deforming material body in an arbitrary manner indicated by Ω* = Y(B, D, W) , where B and D and W are the left Cauchy Green tensor and the stretching tensor and the vorticity tensor, respectively. Then the corotational rate of σ defined by the spin Ω* , i.e., the tensor field σ* = σ+σΩ*—Ω*σ , is objective for every time-differentiable objective Eulerian symmetric tensor field a if and only if the spin tensor Ω* assumes the form Ω* = W + Y(B, D) . where Y(B, D) is an antisymmetric tensor-valued isotropic function. Furthermore, by virtue of certain necessary or reasonable requirements, it is found that a single antisymmetric function of two positive real variables can be introduced to characterize a general class of spin tensors defining objective corotational rates. Accordingly, a general explicit basis-free expression for the latter is established in terms of the left Cauchy-Green tensor B , the vorticity tensor W and the stretching tensor D as well as the introduced antisymmetric function. By choosing several particular forms of the latter, it is shown that all commonly-used spin tensors are incorporated into this general expression in a natural way.

Existence and uniqueness of the integrable-exactly hypoelastic equation\(\mathop \tau \limits^ \circ = \lambda (trD){\rm I} + 2\mu D\) and its significance to finite inelasticity

SummaryIn Eulerian rate type finite inelasticity models postulating the additive decomposition of the stretchingD, such as finite deformation elastoplasticity models, the simple rate equation indicated in the above title is widely used to characterize the elastic response withD replaced by its “elastic” part. In 1984 Simo and Pister (Compt. Meth. Appl. Mech. Engng.46, 201–215) proved that none of such rate equations with several commonly-known stress rates is exactly integrable to deliver an elastic relation, and thus any of them is incompatible with the notion of elasticity. Such incompatibility implies that Eulerian rate type inelasticity theory based on any commonly-known stress rate is self-inconsistent, and thus it is hardly surprising that some aberrant, spurious phenomena such as the so-called shear oscillatory response etc., may be resulted in. Then arises the questions: Whether or not is there a stress rate

The choice of objective rates in finite elastoplasticity: general results on the uniqueness of the logarithmic rate

\mathop {\tau ^* }\limits^\bigcirc

On the simultaneous estimation of model parameters used in constitutive laws for inelastic material behaviour

? The answer for these questions is crucial to achieving rational, self-consistent Eulerian rate type formulations of finite inelasticity models. It seems that there has been no complete, natural and convincing treatment for the foregoing questions until now. It is the main goal of this article to prove the fact: among all possible (infinitely many) objective corotational stress rates and other well-known objective stress rates