Instytut Podstawowych Problemów Techniki

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A class of nonsingular yield conditions depending on three parameters is analyzed for isotropic materials exhibiting strength differential effect and pressure insensitivity. The yield condition can then be expressed in terms of the second and third stress deviator invariants. The convexity requirement is considered and the constraints imposed on the material parameters are discussed in detail. The dual dissipation function is derived in the analytical form. The condition can be applied in the analysis of high strength alloys (such as Inconnel 718) or of shape memory alloys (such as NiTi, NiAl, CuZnGa, or CuAlNi) in order to specify the onset of yield, or of martensitic or austenitic transformation. The conditions can easily be generalized to account for mixed hardening and back stress anisotropy. Some experimental data are provided to verify the proposed conditions.

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Presuming that the incremental free energy is invariant under a change of the Lagrangean finite strain measure and/or the reference configuration Hill’s transformation rules for the basic quantities occurring in mechanics of elastic–plastic solids are recasted in general 3D situation. On this background the invariant incremental plastic work is defined. The basic connections between Hill–Rice theoretical framework and Eckart–Mandel approach, involving the mobile stress-free configuration, are discussed both in generalized coordinates and in the tensorial notation. To this end the selected fundamentals of solid mechanics including the work-conjugacy are recalled. The structure of the updated Lagrangean plastic increment of the total strain is exhibited accounting for the deformation and stress effects due to possible damage and pressure sensitivity of a solid. Special simple approximate relations are derived for the situations when non-dilatational elastic strains are small. The merits of using the logarithmic elastic strain as a state variable are also discussed.

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Taking for granted that the free energy function is invariant under a change of a finite strain measure and/or the reference configuration, Hill's transformation rules for selected fundamental constitutive quantities (such as tangent elastic modulus, plastic increments of total strain and work conjugate stress, the work of work-conjugate stress, the work expended in the plastic part of incremental strain etc.) are derived in a manner different from that of Hill. On this background distinguished by Hill [6] subtle aspects of invariance in mechanics of elastic plastic solids are discussed. It is shown that the plastic part of the increment of elastic strain energy (when taken with reverse sign) defines the true invariant incremental plastic work which in general is not equal to the work expended in the plastic part of the strain increment. It plays the role of a potential for the plastic part of the increment of work-conjugate stress. This fundamental fact has not found proper account in the literature. The analytical interrelations between two apparently different theoretical frameworks, Hill-Rice (fixed reference configuration) and Eckart-Mandel (mobile unloaded configuration) are discussed showing their equivalence. Since the transformation rules are complex in the general 3D case, the first part of the paper illustrates instructively the discussed aspects in a 1D situation (simple tension or simple extension).

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The thermodynamic theory of the pseudoelastic behaviour of SMA is generalized to include new observed effects in isotropic solids. It has been shown that macroscopic eigenstrain due to martensitic phase transitions (p.t.) is a homogeneous function of the stress of order zero, provided that partial thermodynamic equilibrium occurs. The specific form of the Gibbs potential is presented and new conditions for the initiation of p.t. are derived. They are expressed in terms of the temperature, second and third invariants of the stress deviator. The thermostatic properties and phenomenological constants are found for a NiTi alloy using experimental data reported in the literature. The theoretical and experimental results are compared for simple tension compression and pure shear.

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Using the basic principles of continuum thermodynamics the notion of the thermodynamical driving force for coherent phase transformations is introduced. The new representations of thermodynamical driving force are derived. It is shown that thermodynamical driving force reaches an extremum at any dynamically admissible phase amplitude, provided that two phase solids exhibit instantaneous elasticity. The possible interpretation of macroscopic thermodynamical driving force for martensitic transformation is presented.

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In the present paper, Hill's 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.

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The problems of quenching-stress analysis are critically reviewed and an adequate simple theory is discussed. The theory accounts for both plasticity and volumetric changes due to phase transformation accompanying the thermal-hardening of a group of simple steels that are characterized by C-shaped time-temperature-transformation diagrams. The volume dilatation in the absence of stress is assumed to be a linear function of the separate specific volumes and weight fractions of the constituents (pearlite, austenite and martensite). With use of the classical relationships of a formal theory of transformation kinetics, the amounts of pearlite and martensite are expressed in terms of the temperature and the temperature-history. The specific forms of such functions are given. In order to account for the influence of phase transformation on plastic properties, the non-isothermal plastic flow-rule is generalized, and a thermal-hardening parameter is introduced which is identified with the amount of pearlite. Variational principles and bounding inequalities associated with the fundamental rate-problem are considered. As an example, the problem for a rapidly, uniformly-cooled half-space is solved. The variations of the residual stress and the final amount of martensite with distance from the outer surface are given, for several values of the rate-of-cooling. The results suggest that the residual stress vanishes on the plane containing approximately 30–35 per cent of martensite.

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Kinetics of phase transformations in steel during quenching is presented. The transformations form austenite to pearlite and to martensite are characterized as a function of thermal history, and the volumetric change of an element is discussed. A plastic flow rule including such effects of structural changes is proposed in the second part of this paper. A numerical example is shown for a semi-infinite body cooled at the surface from an eutectoid temperature.

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The problem of the transverse collision of a finite circular cylinder (flat-ended metal projectile) of length L and radius R0, moving with the initial velocity V0, with an immobile thin infinite plate, is considered. It is assumed that the plate is in a pure shear state. The one-dimensional waves both in the plate (cylindrical elastic-plastic waves) and in cylinder (plane longitudinal elastic waves) are investigated. Graphs of the peak dimensionless shear stress at the point of impact versus four non-dimensional quantities representing the material and geometrical properties of the plate and the projectile, are presented. These graphs allow the projectile speed just sufficient to initiate ejection of the slug to be estimated.

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This article presents the proof of local and global uniqueness conditions in rate independent coupled thermoplasticity. The mathematical structure of rate equations which account for all coupling effects was discussed in [1] where uniqueness conditions were stated without proof and discussed for some particular cases.

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Rate equations of coupled thermoplasticity are derived assuming that the total strain rate is composed of elastic and plastic terms and accounting for coupling effects such as heat of dissipation, piezocaloric effect, in both mechanical and temperature equations. Local and global uniqueness conditions are presented and its engineering significance is discussed.

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The coupled thermoplastic behavior is studied within the framework of classical thermodynamics. A classification of thermal effects is given and a pattern for theories allowing for a straightforward interpretation of the observed properties of irreversible deformation is presented. A procedure is outlined for constructing a simple non-isothermal theory of thermoplasticity. A suitable single internal parameter is introduced. The Ziegler orthogonality principle is used in deriving the equations regarding the temperature, the stress and the rate of plastic strain. Explicit forms of the governing equations for elastic-plastic materials with an isotropic hardening are derived.

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The constitutive relations of the theory proposed in the first part of the paper are analyzed as to their consistency. The inversion of the obtained rate relations and the uniqueness of the solution to boundary value problems in coupled thermoplasticity are discussed. Simplified rate relations are derived on neglecting certain couplings. The significance of coupling effects in the stability analysis of thermo‐plastic deformations is outlined. The response of structural elements to thermal and loading cycles is described and the bounding theorems of the shakedown analysis are recalled.

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The variational principle and bounding inequalities are stated for infinitesimal rate-independent thermoplasticity. They concern simplified version of rate equations which are obtained by neglecting some coupling effects of lesser importance.

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Stress-rate and strain-rate variational principles are derived for a viscoplastic material characterized by additive elastic and inelastic strain rates. Elastic, viscous and plastic properties are assumed to be temperature dependent but all coupling effects are neglected.

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