| 1. |
- Adolfsson, Erik, et al.
(författare)
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Matrix crack initiation and progression in composite laminates subjected to bending and extension
- 1999
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Ingår i: International Journal of Solids and Structures. - 0020-7683 .- 1879-2146. ; 36:21, s. 3131-3169
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Tidskriftsartikel (refereegranskat)abstract
- An experimental investigation of matrix crack initiation and progression in glass/epoxy laminates of different stacking sequences is presented. The laminates have been loaded in extension and bending, and the degree of damage as function of the load has been recorded. The changes in certain elastic properties caused by the damage were also measured, and are compared to results from a previously developed approximate analytic model. An energy release rate resistance curve is adopted in an attempt to describe the initiation and progression of matrix cracks in the laminates. The amount of cracking is also viewed in relation to the strain transverse to the fibres in the ply under consideration, and the ply stresses at the onset of cracking are calculated. The different damage evolution criteria are compared to the experimental results, and their validity and reliability are discussed. By use of the ply strain transverse to the fibres as a critical parameter for damage evolution, the load-deformation curves of the tested laminates are simulated taking damage progression into account.
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| 2. |
- Adolfsson, Erik, et al.
(författare)
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Thermoelastic properties in combined bending and extension of thin composite laminates with transverse matrix cracks
- 1997
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Ingår i: International Journal of Solids and Structures. - 0020-7683 .- 1879-2146. ; 34:16, s. 2035-2060
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Tidskriftsartikel (refereegranskat)abstract
- Approximate analytic expressions for the thermoelastic properties in combined bending and extension of composite laminates containing transverse matrix cracks are derived. The model covers two-dimensional laminates of arbitrary lay-up sequences. The derived expressions for the compliances and thermal expansion coefficients merely contain ply property data and crack distributions. In order to check the accuracy and reliability of the presented analytic method, some sample cracked geometries were examined by use of the finite element method. A good agreement was found between the numerically and analytically obtained results for all cases under consideration.
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| 3. |
- Christensen, Peter
(författare)
-
A semi-smooth newton method for elasto-plastic contact problems
- 2002
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Ingår i: International Journal of Solids and Structures. - 0020-7683 .- 1879-2146. ; 39:8, s. 2323-2341
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we reformulate the frictional contact problem for elasto-plastic bodies as a set of unconstrained, non-smooth equations. The equations are semi-smooth so that Pang's Newton method for B-differentiable equations can be applied. An algorithm based on this method is described in detail. An example demonstrating the efficiency of the algorithm is presented. © 2002 Elsevier Science Ltd. All rights reserved.
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| 4. |
- Ciavarella, M., et al.
(författare)
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Interaction of thermal contact resistance and frictional heating in thermoelastic instability
- 2003
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Ingår i: International Journal of Solids and Structures. - 0020-7683 .- 1879-2146. ; 40:21, s. 5583-5597
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Tidskriftsartikel (refereegranskat)abstract
- Thermoelastic contact problems can posess non-unique and/or unstable steady-state solutions if there is frictional heating or if there is a pressure-dependent thermal contact resistance at the interface. These two effects have been extensively studied in isolation, but their possible interaction has never been investigated. In this paper, we consider an idealized problem in which a thermoelastic rod slides against a rigid plane with both frictional heating and a contact resistance. For sufficiently low sliding speeds, the results are qualitatively similar to those with no sliding. In particular, there is always an odd number of steady-state solutions, if the steady-state is unique it is stable and if it is non-unique, stable and unstable solutions alternate, with the outlying solutions being stable. However, we identify a sliding speed V0 above which the number of steady states is always even (including zero, implying possible non-existence of a steady-state) and again stable and unstable states alternate. A parallel numerical study shows that for V > V0 there are some initial conditions from which the contact pressure grows without limit in time, whereas for V < V0 the system will always tend to one of the stable steady states. © 2003 Elsevier Ltd. All rights reserved.
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| 5. |
- Giannakopoulos, A. E., et al.
(författare)
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The Stresses Around A Partly Microcracked Hole In Certain Ceramic Materials Under Internal-Pressure.
- 1991
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Ingår i: International Journal of Solids and Structures. - 0020-7683 .- 1879-2146. ; 28:3, s. 329-339
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Tidskriftsartikel (refereegranskat)abstract
- In the present work we examine the stress field around a hole in certain ceramic materials. The hole is under internal pressure in plane strain conditions. The material behavior is initially isotropic linearly elastic and upon loading develops microcracks. The damage is assumed to be anisotropic and microcracking is believed to take place along preferred orientations. A damaged constitutive law, appropriate for the microcracking assumptions, is used to derive exact results for the stress fields in this particular problem. The present analysis may find applications in the experimental investigation of ceramic materials behaving as anisotropically microcracking solids.
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| 6. |
- Girhammar, Ulf Arne, et al.
(författare)
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Dynamic analysis of composite members with interlayer slip
- 1993
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Ingår i: International Journal of Solids and Structures. - : Pergamon Press. - 0020-7683 .- 1879-2146. ; 30:6, s. 797-823
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Tidskriftsartikel (refereegranskat)abstract
- The exact and approximate analysis of composite members with partial interaction and subjected to general dynamic loading are presented. General closed-form solutions for the displacement functions and the various internal actions in the composite elements are presented for both the exact and approximate cases. The solutions reduce to the well-known values for the extreme cases of interlayer connection: non-composite and full composite action, respectively. An approximate solution for the eigenfrequencies in the exact analysis procedure is proposed and evaluated by comparison with the exact solution. These solutions show the effect of interlayer connection on the eigenfrequency. The exact and approximate analysis procedures are applied to simply-supported beams subjected to impulsive and step loadings to illustrate the difference in the solutions obtained by the two procedures. Very good agreement is obtained between the results of exact and approximate analysis of these particular cases
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| 7. |
- Gudmundson, Peter, et al.
(författare)
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An analytic model for thermoelastic properties of composite laminates containing transverse matrix cracks
- 1993
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Ingår i: International Journal of Solids and Structures. - 0020-7683 .- 1879-2146. ; 30:23, s. 3211-3231
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Tidskriftsartikel (refereegranskat)abstract
- An analytical model for the prediction of the thermoelastic properties of composite laminates containing matrix cracks is presented. In particular, transverse matrix cracks with their crack surfaces parallel to the fibre direction and perpendicular to the laminate plane are treated. Two- and three-dimensional laminates of arbitrary layup configurations are covered by the model. The presented expressions for stiffnesses, thermal expansion coefficients, strain contributions from release of residual stresses and local average ply stresses and strains do solely contain known ply property data and matrix crack densities. The key to the model is the judicious use of a known analytical solution for a row of cracks in an infinite isotropic medium. The model has been verified against numerically determined stiffnesses, thermal expansion coefficients and local average ply stresses for matrix cracked angle-ply and cross-ply laminates. Comparisons to experimental data for cross-ply laminates are also presented. It is shown that the present model to a very good accuracy can predict thermoelastic properties of matrix cracked composite laminates at varying matrix crack densities and layup configurations.
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| 8. |
- Ireman, Peter, et al.
(författare)
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A model of damage coupled to wear
- 2003
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Ingår i: International Journal of Solids and Structures. - : Elsevier. - 0020-7683 .- 1879-2146. ; 40:12, s. 2957-2974
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Tidskriftsartikel (refereegranskat)abstract
- The present paper presents a model of damage coupled to wear. The damage model is based on a continuum model including the gradient of the damage variable. Such a model is non-local in the sense that the evolution of damage is governed by a boundary-value problem instead of a local evolution law. Thereby, the well-known mesh-dependency observed for local damage models is removed. Another feature is that the boundary conditions can be used to introduce couplings between bulk damage and processes at the boundary. In this work such a coupling is suggested between bulk damage and wear at the contact interface. The model is regarded as a first attempt to formulate a continuum damage model for studying crack initiation in fretting fatigue. The model is given within a thermodynamic framework, where it is assured that the principles of thermodynamics are satisfied. Furthermore, two variational formulations of the full initial boundary value problem, serving as starting points for finite element discretization, are presented. Finally, preliminary numerical results for a simple one-dimensional example are presented and discussed. It is qualitatively shown how the evolution of damage may influence the wear behaviour and how damage may be initiated by the wear process. © 2003 Elsevier Science Ltd. All rights reserved.
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| 9. |
- Isaksson, Per, et al.
(författare)
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Directional crack path criterion for crack growth in ductile materials subjected to shear and compressive loading under plane strain conditions
- 2003
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Ingår i: International Journal of Solids and Structures. - 0020-7683 .- 1879-2146. ; 40:13-14, s. 3523-3536
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Tidskriftsartikel (refereegranskat)abstract
- A directional crack growth criterion in a compressed elastic perfectly plastic material is considered. The conditions at the crack-tip are evaluated for a straight stationary crack with a small incipient kink. Remote load is a combined hydrostatic pressure and pure shear applied via a boundary layer. Crack surfaces in contact are assumed to develop homogenous Coulomb friction. The crack opening displacement of an extended kink is examined in a finite element analysis to judge the risk of opening mode failure. It has been found that the direction that maximizes the crack opening displacement of an extended kink tip coincides very well with a prediction of the crack growth direction obtained by using a criterion for continued crack growth direction discussed by the authors elsewhere [Int. J. Fract. 108 (2001) 351]. Moreover, the by the model predicted incipient crack growth directions are qualitatively comparable with reported crack paths obtained in ductile materials in a limited number of experiments performed under a combined load of inplane shear and compression.
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| 10. |
- Isaksson, Per, et al.
(författare)
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Mode II crack paths under compression in brittle solids : A theory and experimental comparison
- 2002
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Ingår i: International Journal of Solids and Structures. - 0020-7683 .- 1879-2146. ; 39:8, s. 2281-2297
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Tidskriftsartikel (refereegranskat)abstract
- A study of crack propagation paths in the situation where the crack is suppressed to open during growth due to high compressive forces has been performed. This problem was analyzed theoretically very recently by the authors and is here extended to involve a limited number of illustrative experimental results reported elsewhere in the literature. By analyzing the experimental crack growth patterns, the conclusion is that the model cannot describe the more realistic microscopic failure in detail. Since shear crack growth on the microscale strongly depend on inhomogenities in the material, like cavities, grains or inclusions; the closed crack growth patterns observed are not smooth or free of kinks. Nevertheless, the model show good agreement with the reported experimental observations of the paths of closed macroscopic mode II cracks on samples in brittle materials, induced under overall compression. Failure patterns experimentally observed supports the theory that the growth of mode II cracks under compression in brittle materials follow a propagation path described by a function y ¼ kxb, where b ¼ 3=2. This is strongly supported by the measured values obtained from various experiments. In all the studied experiments, the exponent b was found in the interval [1.43–1.58]. Further, an investigation of the curvature parameter k has been performed and the conclusion is that k does also agree with the simplified model, even though not as good as the exponent b. However, k differs in general <15% from the theoretical value predicted by the model. The process of crack growth is in the simplified model assumed to be controlled by the mode II stress intensity factor KII of the main crack and the difference between the compressive remote normal stress parallel with the crack plane (r111) and the compressive remote normal stress perpendicular to the crack plane (r122).
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| 11. |
- Klisinski, Marek
(författare)
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On constitutive equations for arbitrary stress-strain control in multi-surface plasticity
- 1998
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Ingår i: International Journal of Solids and Structures. - 0020-7683 .- 1879-2146. ; 35:20, s. 2655-2678
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Tidskriftsartikel (refereegranskat)abstract
- The structure of incremental constitutive equations in multi-surface plasticity is discussed with respect to different choices of state and control variables. The state and control variables can combine stresses and strains as long as they are decomposed into energy-conjugate parts. A general uniqueness condition is established for non-associated flow rules and any choice of control variables. Furthermore, proper tangent constitutive matrices are given within each loading/unloading region. The theory is demonstrated in a simple example involving Tresca's yield condition
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| 12. |
- Makela, P., et al.
(författare)
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Orthotropic elastic-plastic material model for paper materials
- 2003
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Ingår i: International Journal of Solids and Structures. - : Elsevier BV. - 0020-7683 .- 1879-2146. ; 40:21, s. 5599-5620
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Tidskriftsartikel (refereegranskat)abstract
- Paper and paperboard generally exhibit anisotropic and non-linear mechanical material behaviour. In this work, the development of an orthotropic elastic plastic constitutive model, suitable for modelling of the material behaviour of paper is presented. The anisotropic material behaviour is introduced into the model by orthotropic elasticity and an isotropic plasticity equivalent transformation tensor. A parabolic stress-strain relation is adopted to describe the hardening of the material. The experimental and numerical procedures for evaluation of the required material parameters for the model are described. Uniaxial tensile testing in three different inplane material directions provides the calibration of the material parameters under plane stress conditions. The numerical implementation of the material model is presented and the model is shown to perform well in agreement with experimentally observed mechanical behaviour of paper.
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| 13. |
- Nilsson, K.-F., et al.
(författare)
-
Delamination buckling and growth for delaminations at different depths in a slender composite panel
- 2001
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Ingår i: International Journal of Solids and Structures. - 0020-7683 .- 1879-2146. ; 38:17, s. 3039-3071
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Tidskriftsartikel (refereegranskat)abstract
- A numerical and experimental investigation for delamination buckling and growth for slender composite panels loaded in compression is presented. The investigated panels consisted of 35 plies in a cross-ply layup with artificially embedded delaminations inserted after three, five or seven plies from the upper surface. The tests clearly and consistently showed that for all delamination depths, delaminated panels failed by delamination growth slightly below the global buckling load of the undamaged panel, whereas the undelaminated panels failed in compression at global buckling. The analysis was done with a finite element based computational model that accounts for contact between delaminated members and fracture mode separation and where crack propagation was simulated with a moving mesh scheme. For all delamination depths, the analysis showed a dramatic increase in the energy release rate when global buckling takes place. Features seen in the tests were captured in the computational analysis. Excellent agreement with tests was found for loads at which delaminated members buckle, the load for onset of delamination growth and the evolution of delamination, e.g., delamination shape and out-of-plane displacements.
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| 14. |
- Nilsson, K.F., et al.
(författare)
-
On transition of delamination growth behaviour for compression loaded composite panels
- 2001
-
Ingår i: International Journal of Solids and Structures. - 0020-7683 .- 1879-2146. ; 38:46-47, s. 8407-8440
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Tidskriftsartikel (refereegranskat)abstract
- This paper presents a combined numerical and experimental study of slender composite panels loaded in compression with artificial delaminations at two different depths. The study was motivated by finite element analyses where this change in delamination depth induced a transition in the direction of delamination growth along with a change in the basic fracture modes and stability. Tests were then carried out to verify the transition in delamination growth. The predicted transitions were to a large extent also seen in the tests. The paper gives an outline of the computational model, which includes contact between delaminated members, calculation of energy release rate with fracture mode separation by an approximate as well as a reliable method for general layups, and moving mesh scheme to account for delamination growth. The experimental work includes monitoring of delamination growth by C-scan and acoustic emission along with a detailed fractographical study. The correlation between experimental observations and computed results are discussed in detail.
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| 15. |
- Runesson, Kenneth, et al.
(författare)
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Characteristics of constitutive relations in soil plasticity for undrained behavior
- 1992
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Ingår i: International Journal of Solids and Structures. - : Elsevier BV. - 0020-7683 .- 1879-2146. ; 29:3, s. 363-380
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Tidskriftsartikel (refereegranskat)abstract
- Elastic-plastic relations which are subjected to the constraint of incompressibilily, that is pertinent to undruined behavior of soils, are assessed and compared with the relations for drained behavior. The tangent relations for mixed control variables (total stress and strain components) are established explicitly. The criterion signalling plastic loading, and elastic unloading, is considered in particular. A criterion for unique response, as expressed in terms of elastic and plastic loading, is formulated as a condition on the particular plastic modulus that corresponds to undrained behavior. Stability is discussed in terms of appropriate critical values of the plastic modulus. It appears that, for a frictional material, dilatancy has a stabilizing effect whereas contractancy may be destabilizing compared to drained behavior.
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| 16. |
- Semb, G., et al.
(författare)
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Stresses and Strains in a Race Track Strap with a radical clearance
- 2002
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Ingår i: International Journal of Solids and Structures. - 0020-7683 .- 1879-2146. ; 39:3, s. 699-708
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Tidskriftsartikel (refereegranskat)abstract
- In certain situations it is desirable to transfer only tensile loads between two points in a structure whilst minimising any stress concentrations. In such circumstances a so-called race track starp can be utilised. The strap consists of two semi-circular elements that transfer the load, from bolts at each end, to two parallel flexible elements joining the whole together. The strap is loaded in tension by means of these bolts. This paper considers the situation where there is clearance between the bolts and the semi-circular elements of the strap. To develop an analytical model, it is assumed that engineering beam theory is applicable, that the influence of the membrane strains can be ignored and that the clearance is small as compared with the bolt and strap radii. It was found that the simple analytical model compared well with both finite element calculations and experiments.
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| 17. |
- Singh, U.K., et al.
(författare)
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Application of a continuum damage model in the finite element simulation of the progressive failure and localization of deformation in brittle rock structures
- 1989
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Ingår i: International Journal of Solids and Structures. - 0020-7683 .- 1879-2146. ; 25:9, s. 1023-1038
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Tidskriftsartikel (refereegranskat)abstract
- A continuum damage model is generalized to simulate both the progressive failure and the formation of localization bands in brittle rock structures under both tensile and compressive loading. The model is implemented into a finite element code to simulate the constitutive behaviour of a number of brittle rock structures which might actually be found in the laboratory or in the field. The effect of closed cracks is included in the calculations. Since, in the present work, finitely extended bodies are considered, the damage in a given direction can no longer be supposed uniformly distributed throughout the body considered
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| 18. |
- Singh, U.K., et al.
(författare)
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Continuum damage model for simulation of the progressive failure of Brittle Rocks
- 1989
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Ingår i: International Journal of Solids and Structures. - : Elsevier BV. - 0020-7683 .- 1879-2146. ; 25:6, s. 647-663
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Tidskriftsartikel (refereegranskat)abstract
- A constitutive model is developed in which a set of continuous field variables, called damage vectors are used to describe the anisotropic response of a brittle solid due to the growth of cracks under general applied loads. The model is tested numerically by studying the response of infinitely extended solids under a number of general plane strain loading conditions
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| 19. |
- Storåkers, Bertil, et al.
(författare)
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Similarity analysis of inelastic contact
- 1997
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Ingår i: International Journal of Solids and Structures. - 0020-7683 .- 1879-2146. ; 34:24, s. 3061-3083
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Tidskriftsartikel (refereegranskat)abstract
- Analysis of mechanical contact of solids is of interest not only regarding a variety of mechanical assemblies but also on a smatter scale such as roughness properties of surfaces and compaction of powder particles. indentation testing is another prominent problem in the context. To analyse the phenomena involved is inherently difficult at application essentially due to the presence of large strains, nonlinear material behaviour, time dependence and moving contact boundaries. Recently, progress has been made, however, to explicitly solve basic boundary value problems especially due to advances in computational techniques. A substantial ingredient which facilitates solution procedures is self-similarity and it is the present purpose to explore in detail the advantages in a general setting when this feature prevails. A viscoplastic framework is laid down for a wide class of constitutive properties where strain-hardening plasticity, creep and also nonlinear elasticity arise as special cases. It is then shown that when surface shapes and material properties are modelled by homogeneous functions, associated boundary value problems posed may be reduced to stationary ones. As a consequence, within Hertzian kinematics, relations between contact impression and regions become independent of loading and time and the connection to loading characteristics does not usually require a full solution of the problem. In particular it is shown that for general head-shapes it proves efficient to use an approach where an intermediate Aat die solution serves as a basic tool also for hereditary materials. An invariant computational procedure based on the intermediate problem is arrived at and decisive results shown to be found by simple cumulative superposition. Illustrations are given analytically for ellipsoidal contact of Newtonian fluids and by detailed computations for spherical indentation of viscoplastic solids for which also universal hardness formulae an proposed. For several bodies in contact it is shown how general results may be extracted from fundamental solutions for a half-space.
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| 20. |
- Strömberg, Niclas, 1968-
(författare)
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A Newton method for three-dimensional fretting problems
- 1999
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Ingår i: International Journal of Solids and Structures. - : Elsevier. - 0020-7683 .- 1879-2146. ; 36:14, s. 2075-2090
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Tidskriftsartikel (refereegranskat)abstract
- The present paper concerns the numerical treatment of fretting problems using a finite element analysis. The governing equations resulting from a formal finite element discretization of an elastic body with a potential contact surface are considered in a quasi-static setting. The constitutive equations of the potential contact surface are Signorini's contact conditions, Coulomb's law of friction and Archard's law of wear. Using a backward Euler time discretization and an approach based on projections, the governing equations are written as an augmented Lagrangian formulation which is implemented and solved using a Newton algorithm for three-dimensional fretting problems of didactic nature. Details concerning the implementation are provided.
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| 21. |
- Strömberg, Niclas, 1968-, et al.
(författare)
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Derivation and analysis of a generalized standard model for contact, friction and wear
- 1996
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Ingår i: International Journal of Solids and Structures. - : Elsevier. - 0020-7683 .- 1879-2146. ; 33:13, s. 1817-1836
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Tidskriftsartikel (refereegranskat)abstract
- A model for mechanical contact including friction, wear and heat generation is proposed. By defining an internal state variable for the wear process, a generalized standard model for contact, friction and wear is derived from the principle of virtual power and the fundamental laws of thermodynamics. Within the frame of the generalized standard model some specific constitutive models are presented. For instance, a free energy corresponding to an extension of Signorini's unilateral contact conditions to include the wear process at the interface and having a linear tangential compliance between the relative tangential displacement and the tangential contact traction is suggested. Furthermore, a dual pseudo-potential with a friction and wear limit criterion in agreement with Coulomb's law of friction Archard's law of wear is given. In order to study existence and uniqueness questions, this pair of free energy and dual pseudo-potential is analysed in a one point elastic quasi-static contact problem with two degrees of freedom and thermal effects neglected. The so-called rate problem is solved.
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| 22. |
- Tan, X.C., et al.
(författare)
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Buckling of drill string under the action of gravity and axial thrust
- 1993
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Ingår i: International Journal of Solids and Structures. - 0020-7683 .- 1879-2146. ; 30:19, s. 2675-2691
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Tidskriftsartikel (refereegranskat)abstract
- A number of different equilibrium helical buckling configurations for a tubing or drill string confined within a cylindrical casing or hole of larger radius and buckled under static compressive forces are determined. Our work is more general and informative than earlier work since the solutions relating the buckling load and the postbuckling configuration are given explicitly for the string of weight and at any inclined positions. A consideration of the state of transient string buckling when the string undergoes sudden change in the helix radius and might lose contact with the hole wall is also proposed.
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| 23. |
- Zang, W.L., et al.
(författare)
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Kinked Cracks In An Anisotropic Plane Modeled By An Integral-Equation Method.
- 1991
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Ingår i: International Journal of Solids and Structures. - : Elsevier BV. - 0020-7683 .- 1879-2146. ; 27:14, s. 1855-1865
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Tidskriftsartikel (refereegranskat)abstract
- A boundary integral method for cracks in an anisotropic material is presented. The method is based on the integral equation for the resultant forces along the cracks. The integral kernels contain only a weak logarithmic singularity, which simplifies the numerical implementation. Crack closure is also taken into account in the numerical formulation. Numerical tests are presented to illustrate the efficiency and the reliability of the proposed method.
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| 24. |
- Åberg, Mats, et al.
(författare)
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Micromechanical modeling of transient waves from matrix cracking and fiber fracture in laminated beams
- 2000
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Ingår i: International Journal of Solids and Structures. - 0020-7683 .- 1879-2146. ; 37:30, s. 4083-4102
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Tidskriftsartikel (refereegranskat)abstract
- A micromechanical model for excitation of waves resulting from fiber fracture and matrix cracking in laminated beams is presented. The source is described as a time dependent displacement discontinuity and the wave propagation in the beam is modeled by a higher-order beam theory. The equations of motion defined by the beam model are then formally solved by employing integral transforms. Asymptotically valid solutions are subsequently found using residue calculus and the stationary phase method. As an example, a [90/0(2)/90] beam with a width to thickness ratio of 10 is considered. The dispersion curves resulting from the beam theory and three-dimensional finite element computations are compared and a maximum frequency for applicability of the beam theory is determined. The time response from fiber fracture and transverse matrix cracking is presented. The applications of the results to the analysis of acoustic emission experiments are also discussed.
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| 25. |
- Östlund, Sören, et al.
(författare)
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Asymptotic crack tip fields for dynamic fracture of linear strain-hardening solids
- 1988
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Ingår i: International Journal of Solids and Structures. - 0020-7683 .- 1879-2146. ; 24:11, s. 1141-1158
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Tidskriftsartikel (refereegranskat)abstract
- Asymptotic crack tip fields, for dynamic crack propagation in an elastic-plastic material, have been calculated. The material is characterized by J2flow theory with linear-strain hardening. The possibility of plastic reloading on the crack flank is taken into account. Numerical results for the strength of the crack tip singularity, the angular positions of elastic unloading and possible plastic reloading regions, and the angular variation of the stress and velocity fields, are presented as functions of the crack tip speed and the ratio between tangent modulus and elastic modulus. Calculations have been performed for crack tip speeds below a certain limit velocity which depends on the tangent modulus and the loading conditions. The different loading modes which have been studied are modes I and II (plane strain and plane stress) and mode III (antiplane strain).
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