Instytut Podstawowych Problemów Techniki

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The main objective of the paper is the investigation of the influence of the anisotrophy and plastic spin effects on criteria for adiabatic shear band localization of plastic deformation. A theory of thermoplasticity is formulated within a framework of the rate-type covariance material structure with a finite set of internal state variables. The theory takes into consideration such effects as plastic non-normality, plastic-induced anisotropy (kinematic hardening), micro-damage mechanism, thermomechanical coupling and plastic spin.

The next objective of the paper is to focus attention on cooperative phenomena in presence of the plastic spin, and the discussion on the influence of synergetic effects on localization criteria. A particular constitutive law for the plastic spin is assumed. The necessary condition for a localized plastic deformation region to be formed is obtained. This condition is accomplished by the assumption that some eigenvalues of the instantaneous adiabatic acoustic tensor vanish. A procedure has been developed which allows us to discuss two separate groups of effects on the localization phenomenon along a shear band. Plastic spin, spatial covariance and kinematic hardening effects are investigated at an isothermal process in an undamaged solid. In the second case, an adiabatic process in a damaged solid is discussed when the spatial covariance terms and the plastic spin are neglected. Here the thermomechanical coupling, micro-damage mechanism and kinematic hardening effects are examined. For both cases, the criteria for adiabatic shear band localization are obtained in an exact analytical form.

Particular attention is focused on the analysis of the following effects: (i) plastic non-normality; (ii) plastic spin; (iii) covariant terms; (iv) plastic strain-induced anisotropy; (v) micro-damage mechanism; (vi) thermomechanical couplings. Cooperative phenomena are considered, and synergetic effects are investigated.

A discussion of the influence of the plastic spin, kinematic hardening and covariant terms on the shear band localization conditions is presented. A numerical estimation of the effects discussed is given.

Słowa kluczowe:

plastic-induced anisotropy, plastic spin, localization, micro-damage, stress triaxiality

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The main objective of the present paper is the development of a viscoplastic regularization procedure valid for an adiabatic dynamic process for multi-slips of single crystals. The next objective is to focus attention on the investigation of instability criteria, and particularly on shear band localization conditions.

To achieve this aim, an analysis of acceleration waves is given, and advantage is taken of the notion of the instantaneous adiabatic acoustic tensor. If zero is an eigenvalue of the acoustic tensor, then the associated discontinuity does not propagate, and one speaks of a stationary discontinuity. This situation is referred to as the ‘strain localization condition’, and corresponds to a loss of hyperbolicity of the dynamical equations. It has been proved that for an, adiabatic process of rate-dependent (elastic-viscoplastic) crystal, the wave speed of discontinuity surface always remains real and different from zero. It means that for this case the initial-value problem is well-posed. However, for an adiabatic process of rate-independent(elastic-plastic) crystal, the wave speed of discontinuity surface can be equal zero. Then the necessary condition for a localized plastic deformation along the shear band to be formed is as follows: the determinant of the instantaneous adiabatic acoustic tensor is equal to zero. This condition for localization is equivalent to that obtained by using the standard bifurcation method. Based on this idea, the conditions for adiabatic shear band localization of plastic deformation have been investigated for single crystals. Particular attention has been focused on the discussion of the influence of thermal expansion, thermal plastic, softening and spatial covariance effects on shear band localization criteria for a planar model of an f.c.c. crystal undergoing symmetric primary-conjugate double slip. The results obtained have been compared with available experimental observations.

Finally, it is noteworthy that the viscoplasticity regularization procedure can be used in the developing of an unconditionally stable numerical integration algorithm for simulation of adiabatic inelastic flow processes in ductile single crystals, cf. [21].

Słowa kluczowe:

shear band localization, viscoplasticity, adiabatic process, crystal slip

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Streszczenie:

The main objective of the paper is the investigation of shear band localization criteria for finite elastic-plastic deformations of a single crystal subjected to an adiabatic process. The next objective is to focus attention on the temperature dependent plastic behaviour of the single crystal considered. A constitutive model is developed within the thermodynamic framework of the rate type covariance constitutive structure i.e. it is invariant with respect to diffeomorphism. To achieve this aim a multiplicative decomposition of the deformation gradient is adopted and the Lie derivative is used to define all objective rates for introduced vectors and tensors. Thermomechanical couplings are investigated and a method is developed which allows us to use the standard bifurcation procedure in the examination of the adiabatic shear band localization. The general evolution equation for the Kirchhoff stress tensor is obtained. The fundamental matrix in this evolution equation describes thermomechanical couplings as well as local lattice deformation and rotation. For the particular elastic properties of the single crystal and for some simplified case of the coupling effects the criteria for adiabatic shear band localization are obtained in their exact analytical form. The influence of two important thermal effects, namely thermal expansion and thermal plastic softening on the criteria of localization is investigated. The similar influence of spatial covariance effects (which arise from the difference between the Lie derivative and the material rate of the Kirchhoff stress tensor) is also examined.

It has been shown that by incorporating the thermomechanical effects and the spatial covariance effects into a constitutive law of the elastic-plastic single crystal, the plastic hardening modulus hcrit at the inception of localization is in fact small but positive.

It has also been proved that this thermomechanical theory of single crystals can describe the misalignment of the shear bands from the active slip systems in the crystal's matrix. The computed critical value of the strain-hardening rate hcrit, as well as the difference between the direction of the macroscopic shear band and the primary slip systems of the single crystal appeared to be in accord with recent experimental observations [cf. Chang and Asaro (1981, Acta Metall.29, 241–257) for Al-Cu single crystals and Spitzig (1981, Acta Metall.29, 1359–1377) for Fe-Ti-Mn single crystals].

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The main objective of the paper is the investigation of shear band localization conditions for finite elastic-plastic rate independent deformations of a damaged solid body subjected to an adiabatic process. For the kind of process considered, the thermal effects may play a dominating role. The next objective of the paper is to focus attention on temperature-dependent plastic behaviour of the body considered. Thermomechanical couplings are investigated, and a method is developed that allows application of the standard bifurcation procedure in examination of the shear band localization criteria when influence of thermomechanical couplings and thermal softening effects, together with hardening and micro-damage effects, are taken into consideration. Particular attention is focused on the coupling phenomena generated by the heat resulting from internal dissipation. A set of the coupled evolution equations for the Kirchhoff stress tensor and for temperature is considered. The assumption that the thermodynamic process is adiabatic permits elimination of the rate of temperature and permits us to obtain the general evolution equation for the Kirchhoff stress tensor. The fundamental matrix in this evolution equation describes thermomechanical couplings. For the particular elastic properties of the material and for some simplified cases of the coupling effects the criteria for shear band localization have been obtained in exact analytical form.

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The main objective of this paper is the investigation of the influence of thermomechanical couplings and thermal softening effects on adiabatic shear band localization criteria for finite rate-independent deformation of an elastic-plastic body. The constitutive equations for thermoelastic-plastic J2-flow theory are formulated within a framework of the rate type covariance structure with internal state variables. Two alternative descriptions are presented. Both constitutive structures formulated are invariant with respect to diffeomorphisms and are materially isomorphic. Particular attention is focused on the coupling phenomena generated by the internal heat resulting from internal dissipation. An identification procedure has been developed which permits the determination of the exact form of the evolution equation for the internal state variable vector. A set of coupled evolution equations for the Kirchhoff stress tensor and for temperature is investigated. The assumption that the thermodynamic process considered is adiabatic permits the elimination of the rate of temperature and gives the fundamental cvolution equation for the Kirchhoff stress tensor. This important result allows the use of the standard bifurcation method in the examination of the adiabatic shear band localization criteria. For the particular clastic properties of the material and for some simplified case of the coupling effects the criteria for adiabatic shear band localization are obtained in exact analytical form. Discussions of the influence of thermomechanical couplings, thermal expansion, thermal plastic softening effects and the covariance terms on the localization criteria are presented.

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Attention is focused on the description of the combined isotropic and kinematic hardening effects in plastic solids. It has been found that making use of the simple geometrical relation permits us to determine the coefficient in the evolution equation describing the kinematic hardening of the Ziegler type in such a way that the evolution law is consistent with the loading criterion and satisfies the time independence requirement. On the other hand this method leaves room for the identification procedure for the material constants based on available experimental results. General constitutive and evolution equations for plastic solids are formulated. The plastic potential is assumed different than the yield criterion. Simplifications for the associated flow rule are also investigated. A new evolution law for the anisotropic hardening is discussed. This law represents the linear combination of the Prager and Ziegler kinematic hardening rules. Application of the theory developed to the description of the plastic behavior of damaged solids is given. A particular example is considered. The discussion and the interpretation of the isotropic and anisotropic hardening moduli are presented.

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The main objective of the paper is the investigation of shear band localization conditions for finite elastic-plastic rate independent deformations of damaged solids. The first part of the paper is devoted to the formulation of the constitutive relations for elastic-plastic solids when isotropic and kinematic hardening effects and the micro-damage process are taken into consideration. The isotropic work-hardening effect is incorporated in the theory directly by defining the work-hardening-softening material function while the kinematic hardening effect and the softening effect generated by the micro-damage process are described by means of the internal state variable method. The second part of the paper aims at the investigation of the localization of plastic deformations. Different effects on the localization phenomenon are investigated. Particular attention is focused on kinematic hardening and micro-damage effects. It has been found that the influence of these two cooperative phenomena on the onset of localization within shear bands has synergetic nature. The results obtained are in good agreement with recent experimental observations.

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Słowa kluczowe:

crystallographic shear, plasticity, work hardening, damaged solids, shear band localization, elastic plastic rate independent deformations, isotropic work hardening, kinematic hardening, porous solid

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Changes in geometry of a rigid-plastic structure at yield resuld either in an increase of the load carrying capacity or in catastrophic collapse limit analysis of incipient plastic motion does not cover the question of stability,for which a geometrically nonlinear theory has to be employed In this paper stability at the yield-point load is considered for determining the slope of load-deflection curves at the onset of yielding is developed. Regions of stable and unstable load combinations are specified on the load interaction curves for portal frames.

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Effects of changes of geometry on the load-carrying capacity are studied. Using the equations of moderately large deflexion shell theory, two methods for estimation of the load-deflexion relations for perfectly plastic shells are discussed. Results for cylindrical pinned and clamped shells are given. Certain experimental results concerning post-yield behaviour of clamped shells are commented upon.

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