Instytut Podstawowych Problemów Techniki

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An optimal design technique is developed for rigid-plastic cylindrical shells subjected to a distributed transverse pressure and a specified axial load. Moderately large deflections are taken into account and a deformation-type theory of plasticity is employed. The optimal design procedure results in a unified approach to optimization in the post-yield range. Necessary conditions for optimality are established by the aid of variational methods of the optimal control theory. Two examples are presented: (1) the optimal layout of rigid circular supports (stiffeners) is found which minimizes the mean deflection; (2) the optimal thickness distribution of a sandwich shell is established which corresponds to the minimum material consumption requiring the deflection of the design coinciding with that of the constant thickness shell.

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Computerized axial tomography is applied to study the evolution of materials deterioration with increasing plastic straining. Tomographic measurements of the X-rays intensity are presented for uniaxially loaded pure aluminium specimens. Various permanent elongations are considered. Observed changes in local properties and their global variation in the scanned slice are related to permanent strains. For reference, the data recorded are reported to values and features concerning an undeformed specimen. Histograms and standard deviations of the quantity measured in the scanner employed are given. From the results of tomographic and mechanical tests, pertinence of the method to nondestructive qualification of the deformed material can be estimated. Further necessary studies regarding radiological properties of deformed metals are outlined when attempting a specification of the relation between tomographic measurements of the X-ray absorption or intensity and density variations due to the straining induced internal damage.

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A method of bounding from below the load-displacement relation for rigid-plastic structures in stable post-yield ranges is discussed. Moderately large displacement theories are considered where strains involve terms nonlinear in stain gradients. The method employs the principle of virtual work, the convexity property of yield loci referred to the undeformed configuration of a structure and uses bounds to the internal dissipation.

An explicit bound is given for plates and an example illustrating application of the method is presented.

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Internal continuum damage of solids is modelled by an arrangement of oriented cracks. The observed overall mechanical response of directionally damaged materials is anisotropic. Differences in elastic, plastic and failure anisotropics are illustrated for a homogenized damaged material and uniaxial stress experiments were performed on continuously damaged alumium alloy sheets. The stress principal directions varied with respect to those of the introduced damage tensor. Experimentally obtained values of material constants for an orthotropic homogenized solid are given. The preliminary tests allow an approach to mathematical modelling of anisotropic continuous-damage mechanical response within the tensor functions theory for two independent tensor variables.

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In elastic-perfectly plastic beams and frames finite zones of yielding develop under the collapse load. The classical deflection analysis assumes that the elastic rigidity reduces locally to zero at the yield hinges only. Such an approach underestimates the deflections at collapse.

A method is proposed in order to evaluate the displacements of elastic-plastic frames when the actual spread of plastic zones is included in the analysis. In comparison with the classical method the developed technique accounts for the flexural stiffness variation between the hinges and due to the partial yielding of cross-sections. The actual extent of plastic zones depends on the bending moment distribution in the structure.

The method contains both the algorithms and program for the deflection evaluation, specified for I-shape cross-sections. The elastic-plastic moment-curvature relations are derived and integrated to obtain additional rotations due to the flexural stiffness variation when the structure transforms in a mechanism. A linear programming program developed earlier is modified to account for the supplementary rotations. The essential features of the method and those of the program application are illustrated purposely in simple examples, showing the order of magnitude of the actual displacements in comparison with the results of the classical deflection analysis of elastic-plastic beams and frames.

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Internal damage due to fissuration results in an overall anisotropic material behaviour. A scalar damage parameter does not allow one to model a direction dependent response of continuously damaging solids. Employing the idea that the scalar damage parameter can be associated with an appropriate strain a damage tensor related to strains is introduced. The overall elastic properties are determined using a simplified form of the sensor function representations and the stress-strain relations regarding the overall response are given. For combined stress states an uncoupled damage theory is derived for materials with no lateral deformability under axial stress. Both brittle-ductile and elastic-brittle cases are studied using circular plates, for the elastic-brittle case only the governing differential equation are presented.

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Impulsively loaded plastic structures deform beyond the limits of applicability of the geometrically linear theory. It was experimentally observed that due to the membrane action actual permanent displacements are smaller than those predicted by the infinitesimal theory. Exact solutions for deformed shapes in the geometrically nonlinear range are not known for anisotropic structures.

The note advances a technique allowing to bound from below the permanent, moderately large deflection at a chosen point of a rigid-plastic, dynamically loaded structure. The method originally developed for isotropic solids and introducing an auxiliary kinematically admissible velocity field allowing to estimate the dissipation due to the nonlinear terms in the strain rates is extended to orthotropic plates and shells.

Lower bounds are obtained to maximum deflections of circular orthotropic plates obeying a piece-wise linear yield criterion when accounting for moderately large displacements. The influence of orthotropy on the permanent deflections is discussed and the results are compared to those of the linear theory. Meaningful differences are noticed, particularly for more intense impulses. Results for a cylindrical shell are also presented.

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Within the tensor functions representations constitutive relations are studied for solids with a directional permeability. The overall anisotropy of an originally isotropie skeleton is due to the arrangement of microchannels and it varies with the deformation induced.

The laws concerning deformation, fluid flow, couplings between fluid and solid motions as well as those regarding evolution of anisotropy are established.

An application of the polynomial representations of tensor functions of scalar, vector and tensor independent variables allows to devise a consistent nonlinear theory of anisotropic consolidation and thus results in the general form of stress-strain-permeability relations.

Anisotropic filtration and equations governing the mechanical response of linearly elastic, anisotropic, porous solids are recovered as special cases.

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Plastic pre-strain induces directional material response. It has been observed that uni-directional pre-strain results in anisotropic creep. Creep tests in various directions with respect to the direction of pre-strain have shown different steady creep rates and times to rupture.

A theory of anisotropic creep is proposed employing the properties of tensor functions. The creep rate in the steady creep range is expressed as a tensor valued tensor function of the stress and pre-strain. In comparison with the Odqvist theory additional material constants appear in the law proposed. The constants are established and predictions from the theory advanced are compared with the results of experiments concerning two pre-strain magnitudes and seven inclinations of the uniaxial creep tests with respect to the pre-strain direction.

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Geometrically nonlinear behavior of rigid-plastic shells is studied. Consistent sets of relations regarding stress and deformations of solids at finite strains and displacements are recalled first. Simplifying assumptions concerning mostly the kinematics are then introduced to arrive at geometrically nonlinear theories of plastic shells. A classification of theories is given in sequence. Influence of large strains on the load surfaces is discussed. Post-yield behavior of shells is studied as the allowance for large displacements modifies the load deformation behavior and affects stability at the yield point. Examples concerning cylindrical shells in the post yield range are given.

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The applicability of the tensor function theory to the development of elaborate and rational constitutive equations of inelasticity is discussed to facilitate the description of the complicated inelastic response of structural components. After a brief review of the representation theory of scalar-valued as well as tensor-valued functions of a set of vector and tensor variables, previous results in this line of approach are first presented. Then, the formulation of some non-classical constitutive equations of inelasticity is discussed in some detail to elucidate the utility and the practical procedures of this theory; a yield criterion and a non-associated flow law of prestrained plastic materials, those of transversely isotropic materials, isotropic and anisotropic hardening theories of creep, and a constitutive equation of anisotropic creep damage in metals. It is shown that the present approach provides the general framework to materialize accurate constitutive equations of inelasticity.

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A criterion of yielding for homogenized materials with an internal structure generated by perforation is developed using the stress and the mixed stress-perforation tensor invariants. For a square pattern of perforation an experimental homogenization procedure is used to determine components of the perforation tensor introduced. A yield condition for the studied materials is proposed and compared with experiments conceived and performed for the purpose.

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Yielding and failure of oriented solids is studied within the framework of tensor representations. Transversely isotropic materials are considered. Using the information available regarding the tensor generators and the set of independent stress and mixed stress-material orientation invariants the general form of constitutive relation for incipient plastic motion of transversely isotropic solid is given and the yield condition is discussed. Specific forms of yield criteria for cohesive materials as well as for materials with internal friction are developed and compared with experimental information. A novel approach to flow and failure of oriented solid is thus explained on example of stratified material, starting with experimental motivation, passing through theoretical development and terminating on comparisons with experiments.

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Flow, hardening and failure of a soft orthotropic rock is studied under isotropic pressure. A theory employing the tensor functions representation is outlined. Experimental data regarding deformation modes, volume changes and yielding are presented. Anisotropic deformation of a stratified rock (diatomite) subjected to a sequence of isotropic pressures is demonstrated in the permanent deformation range. Originally circular cylindrical specimens are shown to deform to elliptic inclined cylinders depending on the orientation of privileged directions of anisotropy to the cylinder generator. Experiments disclose essential features of mechanical anisotropy in the irreversible deformation range.

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Probability of collapse of plates of random plastic moment is considered within the yield-line theory. The unit yield moments in the hinges are assumed statistically independent. To account for combined stress states in a hinge line a weight factor is introduced. Discrete yield patterns are considered and the collapse load at a required level of significance is computed. The proposed simple technique accounts for randomness of mechanical properties of plates while retaining the computational scheme of a deterministic limit analysis.

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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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The concept of an equivalent orthotropic material is used in evaluating the collapse loads for perforated plates. Uniformly loaded circular plates are investigated. Limit pressure solutions are obtained for both the simply supported and rigidly built-in edge conditions. Statically admissible radial and circumferential bending moment fields are found and the associated velocity fields are shown to be kinematically admissible. Numerical results are given in dimensionless form covering arbitrary plate geometries over the entire range of material orthotropy.

These solutions are quite useful in the plastic analysis and design of perforated plates used as pressure vessel heads, tube sheets, reactor core support plates and the like. Perforated plates have considerably higher effective yield strengths when subjected to equi-biaxial loading than when subjected to loading of arbitrary biaxiality and orientation with respect to the penetration pattern. Thus, the equivalent orthotropic plastic material concept is ideally suited to the analysis of such plates. The resulting limit load pressures are substantially lower than the values obtained using isotropic Tresca yield criterion based on the yield strength for equi-biaxial loading. The results are substantially higher than the values obtained using the maximum isotropic Tresca yield condition falling entirely within the orthotropic yield surface.

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Transition of hyperelastic materials into the plastic state is discussed. A condition of yielding is specified and the form of elastic potential is derived for the case of instantaneous yielding of a sphere. The outset of yielding is examined for Huber-Mises and Tresca yield criteria. Elastic-plastic bending of a plate strip is considered.

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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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Behavior of elastic-plastic structures under repetitive and fluctuating loads is considered in this paper. Plastic deformation either stabilizes after a finite number of cycles or continues during the cycling. In the first case the structure is said to have shaken down to the boundary at the loading program. If plastic deformation does not stabilize an elastic-plastic structure becomes unserviceable due to either alternating plasticity when yielding occurs repeatedly in the opposite senses or accumulation of plastic strains and progressive increase of permanent displacements. This paper attempts to survey the shakedown theory, including the accommodation of a structure to the prescribed loading range as well as inadaptation and unserviceability.

The notion of shakedown, incremental collapse, ratchetting and alternating plastic deformation are first illustrated with examples. The fundamental theorems on shakedown and inadaptation are presented next, attention being directed to generalizations of the classical Melan and Koiter theorems. Applications of the theorems are given. Available methods for determining the shakedown range on generating self-equilibrated stress fields are discussed. Special techniques applicable to problems involving both dead and fluctuating loads are invoked. Recent studies accounting for inertia forces, geometrical changes, material hardening, variation of material properties with temperature, displacement assessment, etc. are referred to.

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The problems of irreversible deformation of metallic structures have come to the very focus of interest in nuclear reactor technology.

The paper surveys developments of the theory of thermoplasticity. Starting with the notions of thermodynamics of continua a theory of rheological materials with internal changes implied by plastic deformations is presented. Both a general theory of elastic-viscoplastic materials and its experimental motivation are discussed. Thermoplastic relations at infinitesimal deformations are considered.

The second part concerns thermo-plastic boundary value problems. Solutions and methods of solution are discussed for problems of quenching, thermal shocks, internal heat generation and machine parts design. Thermal cycling and combined thermo-mechanical loading are considered, methods of shakedown analysis being presented.

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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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A method of finding complete limit analysis solutions for orthotropic plates is developed. For a yield condition suitable for reinforced concrete slabs, existence of hyperbolic and parabolic stress regimes is shown. Presence of hyperbolic zones is responsible for differences between the exact and the yield line theory solutions. Differences between isotropic and orthotropic plate formulas for the yield point loads are shown. Examples of various boundary conditions are considered.

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A method of bounding the shakedown domain for shells is proposed. Starting from W.T. Koiter's ‘inadaptation’ theorem and using the consequences following from the associated flow rule for piece-wise linear yield surfaces a procedure is developed to bound from above the shakedown loading programme for a shell. For multi-parameter loadings a time independent interaction surface bounding the shakedown domain from above is obtained whenever the elastic and the limit analysis solutions for the individual loads are known. An example of shakedown interaction curve for a two-parameter loading of a cylindrical shell is given.

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Une forme générale des relations contrainte-vitesse de déformation est étudiée pour un mouvement non visqueux. Le tenseur contrainte est représenté par une fonction tensorielle isotrope du tenseur vitesse de déformation: le nombre et la forme des fonctions indépendantes décrivant le matériau sont déterminés.

Le principe de la vitesse de dissipation maximale est employé pour obtenir des restrictions supplémentaires pour la relation constitutive. De plus, ce principe permet d'expliquer les contradictions provenant de la forme standard de la loi d'écoulement du potentiel plastique appliquée aux sols. Des formes particulières des relations tensorielles linéaires, dans le cas d'incompressibilité et de dissipation constante sont discutés.

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The constitutive laws of the limit analysis of plates are formulated directly in terms of stress resultants and the corresponding kinematical quantities. General tensorial relations for two-dimensional continua are studied in order to disclose the form of a constitutive equation for plates at collapse. The derivation is based on the requirement that the constitutive equation be homogeneous of degree zero in time. Yield locus and the general form of flow rule are derived. The plastic potential flow rule as well as some other satisfying the material stability postulate follow as particular cases. Examples of compatible collapse modes are given.

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Complete solutions of plastic bending of rotationally symmetric plates are presented in the case of non-homogeneity specific to polar nets of extensible reinforcement. The results are compared with those of homogeneous plates, indicating the differences in collapse loads, collapse modes and stress fields. Optimum design of fibre-reinforced plates is studied.

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The general tensorially linear relation between the stress and the strain rate tensors is studied in order to specify the independent material functions describing the kinematics and the bearing capacity of isotropic soils. It is found that the plastic dilatation remains indeterminate at linear yield loci. A general tensorially linear form of the flow law is obtained and its relation to the associated flow law is indicated.

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It is known from experiments that the actual load-carrying capacities of plates are larger than those predicted by the limit analysis theory. The behavior of plastic structures beyond the bending collapse load is influenced by the changes in geometry of a structure during the process of plastic deformation. In the present paper, an investigation of the load-deflection relationship for simply supported rectangular reinforced concrete plates is presented. The tensile membrane action is found to be localized in zones of bending yield hinges. The zones of pure membrane response consequently develop as the load increases. The paper presents a kinematical method of analysis of plastic plates beyond the bending collapse load. Kinematically admissible collapse modes are studied and the associated dissipation functions are derived. Load-deflection relationships are obtained for various yield patterns. The theoretical results are compared with experimental data following from model tests.

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Finite plastic expansion of a thick-walled spherical vessel is studied in presence of neutron radiation. Due to irradiation the yield stress and the hardening modulus vary across the shell wall. The pressure-expansion relation is obtained in case of combined linear hardening and linear attenuation.

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A hardening rule accounting for anisotropy produced by plastic deformations is developed. Transition from an initially isotropic material into an anisotropic one is studied. The proposed rule accounts for translation, rotation and expansion of the initial yield locus.

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Using the strain-mapping method for the Tresca yield condition, the yield surface is derived for a cylindrical shell with a wall reinforced by longitudinal ribs on one side. Results are given for the case where the axial load is zero. As a sample problem utilizing this surface and as an appropriate method for solving nonlinear equations, the solution of a cantilever shell under constant pressure is obtained.

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The relationship between limit analysis and yield-line analysis is investigated. Attention is restricted to simply supported, isotropic slabs subjected to single-point loadings. It is found that conventional yield-line analyses quite often give substantial overestimates of the carrying capacity. A general method is formulated for finding the yield-point load, and various examples are considered.

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The yield condition for a rigid-plastic circular cylindrical shell subjected to axially symmetric loading is described in terms of the circumferential hoop stress n and the axial bending moment m. Different yield conditions are obtained depending on whether the shell is of homogeneous or ideal sandwich construction, and on whether the shell material satisfies Tresca's or Mises' yield condition. Previous work has been limited to Tresca's yield condition.

The example of a long shell subjected to a ring of pressure at its midpoint is considered for each of the four shell yield conditions as well as for a simple “limited interaction” approximation. Complete stress and velocity fields are determined, the limit load is found, and the extent of the plastic region is determined. The results are assessed in terms of the theorems of limit analysis.

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