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Matematyczne modelowanie mechanicznych właściwości materiałów

Wegner T.

Biuletyn Wojskowej Akademii Technicznej

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2005

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Vol. 54, nr 12

5-51

PL

W pracy rozważono zagadnienia dyssypacji energii oraz stateczności równowagi wewnętrznej w odkształconym materiale, które są ściśle związane z problematyką wytrzymałościową. Właściwości materiału w procesie monotonicznego obciążania aproksymowano nieliniowym modelem matematycznym o stałym module odkształcenia objętościowego i zmiennym module odkształcenia postaciowego. Zastosowane w modelu stałe materiałowe wyznaczono w statycznej próbie rozciągania. Wykorzystując pojęcie gęstości energii odkształcenia oraz definicję stateczności, sformułowano kryterium niestateczności stanu odkształcenia materiału. Badając wypukłość funkcji energii odkształcenia postaciowego, sformułowano nowy warunek plastycznego płynięcia, który wierniej opisuje rzeczywiste zachowanie materiału w złożonym stanie naprężenia, na co wskazują badania eksperymentalne wykonane przez różnych autorów.

EN

The questions of energy dissipation and stability of inner equilibrium in a deformed material, closely related to the strength problems were considered. The material properties during the process of monotonic loading were approximated with the use of a nonlinear mathematical model of constant volumetric modulus of elasticity and varying non-dilatational modulus. Used in the model, material constants were obtained in a static tensile test. The instability criterion of strain state for the material was formulated with reference to the concept of density of strain energy and definition of stability. Examination of convexity of the function of non-dilatational strain energy enabled formulation a new plastic yield criterion. It reliably describes an actual behaviour of the material in a complex strength condition, which is evidenced by the experimental studies carried out by various authors.

2

Constitutive relations for dynamic material instability at finite deformation

Beda P. B.

Archives of Mechanics

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2001

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Vol. 53, nr 4-5

319-331

EN

This paper aims to present a mathematically consistent formulation of the second gradient dependence in the constitutive equations for material instability phenomena in case of finite deformations. Thus the set of fundamental equations of the solid continuum (the kinematic equations, the Cauchy equations of motion and the constitutive equations) should also be written for finite deformations. Two basic properties are required: the existence and regular propagation of waves and the generic behavior at the loss of stability. Firstly, the wave dynamics is studied. To encounter the second gradient effects, we should use the third order waves here. Secondly, the system of fundamental equations completed with initial and boundary value conditions forms a dynamical system. Then, identifying material stability with Lapunov stability of a state of the continuus body, the loss of stability should be one of the two basic types of instabilities of dynamical systems: a static or a dynamic bifurcation. These instability modes should be strictly different for a generic dynamical system.

3

On material and geometrical instabilities in finite elasticity and elastoplasticity

Reese S.

Archives of Mechanics

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2000

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Vol. 52, nr 6

969-999

EN

Material instability phenomena arise in homogeneous stress states if nonlinear stress-strain relations are considered. The stability behaviour is investigated by looking at the Gateaux derivative of the first Piola-Kirchhoff stress tensor in the direction of the deformation gradient. This requires to solve a nine-dimensional matrix eigenvalue problem. In the present contribution, it is shown that material instabilities can be clearly differentiated from instabilities of geometrical character. The latter aspect is especially important for the design of new materials, since unstable solution paths under common loading conditions are not desirable. Geometrical instabilities, however, can usually be avoided by choosing appropriate boundary conditions. The derivation in this work leads to a simple stability criterion which allows to describe the stability behaviour of many materials in a very general context.

4

Material instability in the tensile response of short-fibre-reinforced quasi-brittle composites

Wang J., Karihaloo B.

Archives of Mechanics

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2000

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Vol. 52, nr 4-5

839-855

EN

This paper gives the conditions for the onset of instability in the tensile response of short-fibre-reinforced quasi-brittle composites whose deformation is characterised by multiple cracking and localisation. First, the tensile stress-strain relation is established analytically for a body containing multiple bridged microcracks. The material instability is examined using the classical bifurcation criterion, with an emphasis on the role played by fibre bridging in the macroscopic instability. It is found that while the microscopic instability in the bridging traction plays a major role in the macroscopic instability of the composite, it is the level of damage in the matrix that determines when the macroscopic instability is induced by the bridging instability. The satisfaction of the classical bifurcation criterion is identified with several failure modes, depending on the degree of damage in the matrix.

5

Dynamics of viscoelastic model with non-monotone flow curve

Marin A., Balan C.

Applied Mechanics and Engineering

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1999

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Vol. 4, no spec.

21-26

EN

The paper is concerned with the investigation of the transient flow regime in the start-up of the Couette simple shear motion of a viscoelastic model with non-monotone flow curve, as a function of Reynolds and Weissenberg numbers. The present paper is particularly focused in establishing a numerical procedure for the problem under investigation, based on the integral expression of the shear stress in viscometric motions, as a function of the deformation history.