phenomenological theory Research Papers

A reformulation of strain gradient plasticity

2001, Journal of the Mechanics and Physics of Solids

A class of phenomenological strain gradient plasticity theories is formulated to accommodate more than one material length parameter. The objective is a generalization of the classical J2 ow theory of plasticity to account for strain... more

Fig. 1. Distribution of plastic shear strain across the upper half of the sheared layer at U/(e9L) = 10 for various values of 7/L where / is defined for each of the formulations in (36). The tensile stress—stress strain curve of the material is given by (32) with VN =0.2 and v=0.3. The calculation employs 50 elements across the half-width of the layer and reaches U/(g9L) = 10 in 100 incremental steps.

Fig. 2. Effect of the material length parameter 7 on the overall relation between the shear traction and the shearing displacement for the elastic-plastic layer (NV = 0.2 and v = 0.3).

Fig. 3. Torque versus twist based on the general formulation for a solid cylindrical wire of radius R with N =0.2 and v=0.3. The length parameter 72 is the most important in determining the size effect in wire torsion; 7; has very little influence, and 73 does not enter the problem. The calculation employs 100 elements from the center of the wire to its free surface and it reaches «R/e9 = 50 in 500 incremental steps. (Elastic compressibility of the wire does not play a role in the solution, however the value of Poisson’s ratio enters into the normalization used when evaluating G/E.)

Fig. 4. Torque versus twist based on the one-parameter formulation for a solid wire of radius R with N =0.2 and v = 0.3. The calculation details are similar to those stated in Fig. 3.

Fig. 5. Remote stress as a function of normalized volume expansion based on the general formulation for a spherical void subject to hydrostatic tension, goo, at infinity with N =0.2 and elastic incompressibility. The initial void radius is R. The length parameter 7; is the most important in determining the strain gradient effect in void growth; 73 has very little influence, and 7 does not enter the problem. The calculation maps the region exterior to the void onto (0,1) and employs 100 elements in the mapped region. The calculation attains (AV/é9 )/Vo = 50 in 500 incremental steps.

Fig. 6. Remote stress as a function of normalized volume expansion based on the one-parameter formulation for a spherical void subject to hydrostatic tension, oo, at infinity with N =0.2 and elastic incompressibility.

Fig. 7. Torque versus twist based on the deformation theory of Fleck and Hutchinson (1997) for a solid wire of radius R with N = 0.2 and v =0.3. These results can be compared with the results computed using the new formulation in Fig. 3. (Elastic compressibility of the wire does not play a role in the solution, however the value of Poisson’s ratio enters into the normalization used when evaluating G/E.)

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Modeling of enhanced electrocaloric effect above the Curie temperature in relaxor ferroelectrics

by Yuping SHI

2011, Acta Materialia

The electrocaloric (EC) effect is promising in realizing solid state refrigeration, which requires EC materials possessing pronounced pyroelectric effect over a broad temperature range. Pauli's master equation is adopted to investigate... more

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Settling velocities of particulate systems: 12

by Fernando Concha

2001, International Journal of Mineral Processing

In this contribution we show how the phenomenological theory of sedimentation-consolidation processes can be extended to the presence of centrifugal field. The modelling starts from the basic mass and linear momentum balances for the... more

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Theoretical Temperature-Pressure Phase Diagram for {N(CH 3 ) 4 } 2 ZnCl 4

by grigorii kessenikh

2000, Journal of the Physical Society of Japan

A phenomenological approach for construction theoretical phase diagrams with multiple phase transitions including incommensurate transitions is developed for the case when a special triple point, a Lifshitz-type point, is assumed to be... more

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Creep Analysis of Thin-Walled Structures

by Holm Altenbach

2002, ZAMM

Ingenieurwerkstoffe neigen bei erh€ o ohten Temperaturen zus€ a atzlich zum elastischen Verhalten zum Kriechen oder, mit anderen Worten, man kann neben einer spontanen Werkstoffreaktion auch eine zeitabh€ a angige beobachten. Unter Ber€ u... more

Ingenieurwerkstoffe neigen bei erh€ o ohten Temperaturen zus€ a atzlich zum elastischen Verhalten zum Kriechen oder, mit anderen Worten, man kann neben einer spontanen Werkstoffreaktion auch eine zeitabh€ a angige beobachten. Unter Ber€ u ucksichtigung der wachsenden Sicherheitsanforderungen f€ u ur Kraftwerke, Flugzeugkomponenten, chemischen Anlagen usw. ist das zeitabh€ a angige Werkstoffverhalten in den Konstruktionsprozess einzubeziehen. Zahlreiche Bauteile, die unter Kriechbedingungen eingesetzt werden, k€ o onnen als d€ u unnwandig klassifiziert werden. Ihre Analyse ist dann mit drei Problemen verbunden: mit der Auswahl eines geeigneten Werkstoffmodells, mit der Auswahl eines ad€ a aquaten strukturmechanischen Modells und mit der Auswahl eines geeigneten numerischen L€ o osungsverfahrens. Dabei besteht eine enge Verbindung zwischen den drei Problemen. Beispielsweise hat die Auswahl des strukturmechanischen Modells (Balken, Platte, Schale usw.) signifikanten Einfluss auf den numerischen Aufwand. Andererseits wird die Berechnungsgenauigkeit durch das Materialverhaltensmodell beeinflusst, wenn verschiedene Effekte einbezogen oder vernachl€ a assigt werden. Die Kriechmechanik als Zweig der Festk€ o orpermechanik hat eine mehr als einhundertj€ a ahrige Geschichte. Nach einer kurzen Diskussion der geschichtlichen Entwicklung sowie der Anf€ u uhrung ausgew€ a ahlter Referenzliteratur werden zwei Modelle zur Beschreibung des Kriechverhaltens eingef€ u uhrt. Das erste Modell ist dadurch gekennzeichnet, dass die Art des Spannungszustandes keine Rolle bei der Beschreibung des Materialverhaltens spielt, w€ a ahrend in die Formulierung des zweiten Modells eine derartige Abh€ a angigkeit eingeht. Ein typisches Beispiel f€ u ur das zweite Modell ist ein unterschiedliches Zugund Druckverhalten. Dieses kann beispielsweise im Terti€ a arkriechbereich vorausgesetzt werden, da eine Sch€ a adigung durch Zugbedingungen gef€ o ordert wird. Wenn man Druckbedingungen antrifft, k€ o onnen Kriechverzerrungen auftreten, der Sch€ a adigungszustand ist jedoch teilweise eingefroren (u.a. kein Auftreten neuer Hohlr€ a aume). Der letzte Teil ist strukturmechanischen Modellen gewidmet. F€ u ur d€ u unnwandige Strukturen werden in der Regel alle Annahmen in € U Ubereinstimmung mit der ,,D€ u unne'' der Struktur getroffen. Damit kann man beispielsweise Hypothesen f€ u ur die Spannungen, Verzerrungen und/oder Verschiebungen in Dickenrichtung einf€ u uhren, und es werden zu l€ o osende Gleichungen abgeleitet, die bez€ u uglich der Dimension reduziert sind (statt dreidimensionaler Feldgleichungen hat man ein System von zwei-oder eindimensionalen Feldgleichungen). € U Uber die Korrektheit und Genauigkeit dieses Konzepts wird berichtet.

Fig. 1. Principles of modelling the creep behavior — microstructural and phenomenological Both the microstructural and the phenomenological description of the creep behavior can be applied. The differ- ent size level yields various explanations of creep effects. Starting from the microstructure the nucleation and the growth of voids, cavities, etc. are the main observations. Finally, coalescence plays an important role for the lifetime. All these effects are ruled by the local deformation state and the extension of the results based on the micro-mechan- ical effects is connected with some difficulties (for instance, the choice of the representive volume element or the homo- genization technique) discussed partly in [84, 108, 109]. The macroscopic modelling ignores the local changes and the describing of the creep behavior equations is based on macroscopic observations like creep curves (Fig. 1, right). The predictions of the averaged variables (stresses, strains, etc.) are accurate enough for engineering applications, but the predictions of failure states, of the lifetime, ... are sometimes connected with difficulties.

one establishes a stress dependence of the creep, and a better modelling for materials with significant tertiary creep is possible [87]. Summarizing all equations describing the individual mechanisms HAYHURST and co-workers [47, 58] have been formulated a so-called mechanism-based multi-axial creep model reflecting all three creep stages: and N=1 for o; > 0, otherwise N = 0. D is a material constant. The model is proposed for the description of th uniaxial and the multi-axial creep behavior. In contrast to the classical, purely phenomenological model of the KACHA- NOV-RABOTNOV type [55, 90] this model allows one to take into account more mechanisms (hardening, damage, age ing). As shown in [17] the mechanism-based models must be applied carefully since the results of creep estimations fo: plates and shells based on the classical and the mechanism-based models are only similar during primary creep. The lifetime predictions based on both models are not in a satisfying agreement. This was at first an unexpected result because the uniaxial creep curves approximations are similar for both models. Detailed investigations have shown that the multi-axial stress state was the reason for the disagreements. For the material model we have to use the full three: dimensional stress state representation, whereas for the plate or shell model the stress state is represented in a reducec (two-dimensional) form. These effects are known also in other cases [9].

The creep is influenced by hardening and damaging processes and based on the identified secondary creep equa- tion. The equations for primary and tertiary creep can be established by extension of the secondary creep equations. The identification can be realized by analogy using additional experimental results. This approach is presented in several textbooks, for instance [62, 63, 90, 103, 104].

Let us assume that the material behavior is isotropic (that means initially isotropic and isotropic during the whole lifetime/exploitation period). The creep behavior can be presented by several phenomenological theories discussed above. The starting point here is the uniaxial state (constitutive) equation for secondary creep — the Norton’s law, mostly used in practice, which can be presented in the form of a power law

can be defined. a, 6, y are curve fitting parameters (influence or weight of o;) and w,,..., 4g denote material charac- teristics. In [45, 61] coefficients similar to the introduced a, B, and y are used for the characterization of different failure mechanisms.

Fig. 7. Creep curves for celluloid loaded by uniaxial tension load and hydrostatic pressure, T = 338 K, oy,, = 11.5 MPa Fig. 5. Creep curves for austenitic steel EI-257, T = 837 K

Fig. 9. Influence of pure torsion on the volumetric creep: HD-polyethylen, T = 313 K

Fig. 6. Creep curves for HD-polyethylen, T = 77 K

Fig. 10. Volumetric creep in dependence on the kind of loading: Teflon, T = 298 K, o,,, = 8.6 MPa Fig. 8. Compressibility in dependence on the hydrostatic pres- sure p: Teflon, 1: p = 90 MPa, 2: p = 50 MPa, 3: p = 40 MPa, 4: p = 30 MPa, 5: p = 20 MPa, 6: p= 10 MPa

The following discussions are limited by the use of the approximation the uniaxial secondary creep with the help of Norton’s law. Note that the creep law (4) contains 6 parameters uw; and they must be identified by tests. The question what kind of tests one has to perform is open and depends on the experimental equipment available for providing tests and the possibilities to find closed solutions for the creep mechanics problems similar to the tests. Re carr aeampnila lat we enoonck fhe Lallausie eran facke«

Now we consider special cases of the generalized creep law with 3 parameters. It is easy to show that we can atroduce different variants of 3-parameter creep equations:

‘ig. 12. Solutions for a Bernoulli beam based on the Ritz method using polynomial functions: a) time variation of maximum deflec- ion, b) time variation of maximum stress, 1: approximation using elastic deflection function, 2: N = 1, 3: N= 2,4: N=8 In the following we confirm these estimations and compare the solutions with solutions by the Ritz method constructing some convergent test solutions. The Ritz procedure is presented in [9]. As an example we consider a simply supported beam with a rectangular cross section. For the calculation we set ¢g=60N/mm, /= 1000 mn, h= 80mm, b= 30mm and use the creep-damage material model (18) modified with time-hardening function which allows the description of the primary creep:

Fig. 13. Convergence of time dependent solution for a Bernoulli beam using polynomial functions: a) time variation of maximum deflection, b) time variation of maximum stress, 1: N = 1, 2: N = 2, 3: N=3, 4: N=5,5: N=7,6:N=8

Fig. 14 Solutions for a Bernoulli beam using shell elements SHELL 43: a) time variation of maximum deflection, b) time variation of maximum stress, 1: 8 elements, 2: 10 elements, 3: 20 elements, 4: 40 elements, 5: 80 elements, 6: 200 elements For testing the material law implemented into the ANSYS finite element code we introduce the beam considered below as the previous example. For the meshing we use the 4-node shell element SHELL 43. This element is based on

the Reissner-Mindlin type shell theory and contains five integration points through the thickness. Fig. 14 shows the solutions with respect to different meshes. The convergent solution (case 6) has been obtained with 200 elements. This solution has a very good agreement with the solution obtained using the Ritz method, Fig. 13. Similarly to the sensitiv- ity of the solution to the degree of the polynomial discussed below, the mesh sensitivity of the finite element solution can be observed. The mesh justified for the elastic solution (case 1) is not fine enough for an accurate prediction of the time-dependent mechanical behaviour of the beam. The solutions in the cases 8—6 are approximately the same in the steady state and slightly differ in the tertiary creep stage. From the presented creep-damage calculations one can conclude that convergent solutions for the steady state creep can provide adequate (at least qualitatively correct) solutions within the framework of continuum damage mechanics.

Fig. 16. Time dependent stresses for a beam: a) positive stress on the bottom side, b) negative stress on the top side, curve symbol see Fig. 15 Fig. 15 shows the time-dependent solutions for the maximum deflection obtained with shell elements, plan stress elements, and using the Ritz method. It can be seen that if the sensitivity of material damage to the kind o: stress state is taken into account the results obtained by shell and plane elements differ substantionally. For compar- ison, two convergent solutions based on the Ritz method applied to a beam with the Bernoulli assumptions ar¢ plotted. The first solution is obtained with 5 Gauss points for the thickness integration (curves 3), and the second witl 9 Gauss points (curves 4). Both solutions are obtained with the same number of shape functions. The solution witl 9 Gauss points is in a better agreement with the plane stress finite element solution which is seen particularly on the time variation of maximum negative stress, Fig. 16, b). The solution with 5 Gauss points agrees better with the solu tion with the shell-type finite elements. Since the element SHELL 43 contains 5 Gauss points for the thickness integra-

Fig. 17. Creep responses for various stress states: a) responses by tension, torsion, and compression, b) responses by combined tension (compression) and torsion With respect to the analysis of thin-walled structures, two possibilities connected with different analytical and numerical effort are available for the structural analysis. The first one is the use of general three-dimensional equa- tions and solid finite elements. The second one is, for instance, the application of shell equations and shell based elements usually of the Reissner-Mindlin type. The solid concept can be considered as a general and in many cases more accurate for structural analysis, particularly by studying damage effects. However, the numerical effort for the approximate solving of three-dimensional equations is not comparable with the effort applying theories based on dimension reduced equations. In this case we have to assume some hypotheses which are the bases of the formula- tion of two- or one-dimensional equations. It is well-established that the results of calculations based on dimension- reduced equations in some cases yields over- or underestimations and the distributions of the mechanical field quan- tities are partly incorrect and far from the reality. For example, the transverse shear stresses ignored in the classical plate or beam theories result in an incorrect estimate of the strain field. One possibility to improve the results is the use of shear correction factors which are constant and material independent in the case of homogeneous plates or beams. Therefore, such an approach is applied in finite element programs since Reissner-Mindlin elements are more

According to the strain-displacement equations (30) the first term on the right side in eq. (31) can be transformed a follows: With these approximations for the transverse shear and normal stresses the underlined terms in eq. (32) are vanishing, and the remaining terms can be written after some manipulations as

Fig. 18. Time dependent solutions of a clamped beam: a) maximum deflection vs. time, b) shear correction factor vs. time, 1: Bernoul- li-Euler beam theory, 2: first order shear deformation theory with parabolic shear stress distribution, 3: first order shear deformation theory with modified shear stress, 4: solution using the ANSYS code with PLANE 42 elements

edge zones and the middle of the beam (bottom side). All these regions are in tension and the damage evolution is stimulated. The non-symmetric behavior is a result of the damage law depending on the stress state.

Table 1: Creep behavior of PVC after 100 hours (thin-walled tubes under inner pressure and axial force) — compari- son of experimental results and theoretical predictions based on 3-parameter models Table 2: Creep behavior of the aluminium alloy AK4-1T (thin-walled tubes under inner pressure and torsion at 473 K) — compar- ison of experimental results and theoretical predictions based on 3-parameter models

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