Showing papers on "Linear elasticity published in 1975"
Book•
01 Jan 1975
TL;DR: The Finite Element Method as discussed by the authors is a method to meet the Finite Elements Method of Linear Elasticity Theory (LETI) and is used in many of the problems of mesh generation.
Abstract: PART I. Meet the Finite Element Method. The Direct Approach: A Physical Interpretation. The Mathematical Approach: A Variational Interpretation. The Mathematical Approach: A Generalized Interpretation. Elements and Interpolation Functions. PART II. Elasticity Problems. General Field Problems. Heat Transfer Problems. Fluid Mechanics Problems. Boundary Conditions, Mesh Generation, and Other Practical Considerations. Appendix A: Matrices. Appendix B: Variational Calculus. Appendix C: Basic Equations from Linear Elasticity Theory. Appendix D: Basic Equations from Fluid Mechanics. Appendix E: Basic Equations from Heat Transfer. References. Index.
1,497 citations
TL;DR: In this paper, stress intensity calculations are presented for cases of combined tensile and shear loading for a linear elastic material using functions of a complex variable, and a theory is developed to determine the direction of maximum energy release rate.
Abstract: Stress intensity calculations are presented for cases of combined tensile and shear loading for a linear elastic material. Using functions of a complex variable, a theory is developed to determine the direction of maximum energy release rate. A finite element method using virtual crack extensions is also used to determine the energy release rate for crack extensions in various directions and in particular that which gives the maximum energy release rate. Except when shear is more significant than tension, these results give good agreement with available experimental evidence. When shear is most significant, plasticity effects are probably becoming important, thereby invalidating the results of any linear theory. However, the results may still be used to determine K
I and K
II numerically from virtual crack extension calculations of J
1 and J
2 for general two-dimensional geometries.
184 citations
TL;DR: In this article, the two-dimensional, plane stress problem of linear elasticity is analyzed within a state space framework, where the medium considered is homogeneous and isotropic Vlasov's mixed formulation of elasticity.
Abstract: The two-dimensional, plane stress problem of linear elasticity is analyzed within a state space framework The medium considered is homogeneous and isotropic Vlasov's mixed formulation of elasticity is used throughout The field equations are derived in closed form, thus avoiding Vlasov's intermediate infinite series solution Finally, all the properties of the transfer matrix are shown to follow directly from embedding the problem into a state space setting
111 citations
TL;DR: In this article, a numerical solution method was described for the determination of oil film shape and film pressure in the lubricated contact between an elastic sphere rolling on an elastic plane with steady loading and an isothermal film.
Abstract: A numerical solution method is described for the determination of oil film shape and film pressure in the lubricated contact between an elastic sphere rolling on an elastic plane. Steady loading and an isothermal film are assumed. The Reynolds equation (governing film pressure) and the elasticity equation (governing deformation) are solved simultaneously for a lubricant with the pressure-viscosity characteristic $\eta $ = $\eta $$_{0}$e$^{\alpha p}$. A treatment of the elasticity equation is described such that the deformation matrix is sufficiently compact. To give generality to the solution a set of results was subjected to multiple regression, which indicated that the influence of load on film thickness was very small. The regression results compare well with published data. The effect of restricting the amount of lubricant to the contact was also studied. The computed film shapes under such 'starved' conditions were found to be very close to those found by optical interferometry.
106 citations
TL;DR: In this article, an assumed displacement hybrid finite-element procedure was developed for treating a general class of problems involving mixed-mode behavior of cracks is used to solve some two-dimensional, fracture mechanics problems involving rectilinear-anisotropic materials.
Abstract: An assumed displacement hybrid finite-element procedure developed for treating a general class of problems involving mixed-mode behavior of cracks is used to solve some two-dimensional, fracture mechanics problems involving rectilinear-anisotropic materials. This finite-element program uses four "singular" elements which surround the crack tip and "regular" elements which occupy the remaining region. The singular element has a built-in displacement field of the r type with the two modes of stress intensity factors, K 1 and K I I , as unknowns. Displacement compatibility between singular and regular elements is also maintained. Isoparametric transformations are used to derive the stiffness matrix of quadrilateral curved elements. Rectilinear anisotropic, nonhomogeneous, but linear elastic, material properties are considered. The program was checked out by analyzing a bimaterial tension plate with an eccentric crack and a centrally-cracked orthotropic tension plate. The results thus obtained agreed well with those by Erdogan and Biricikoglu, and Bowie and Freese, respectively. The program was then used to analyze two fracture test specimens for which analytical solutions do not exist. The first specimen was the doubly edge-notched tension plate with material principal directions oriented 0°-90° or ′45° to the geometric axes of symmetry and with varying crack length. The second specimen was the three-point bend specimen with material principal directions oriented 0°-90° to the geometric axes of symmetry. Finally, an orthotropic tension plate with an oblique center crack was analyzed. Finite-element solutions of most of these problems do not seem to have appeared in prior literature.
40 citations
TL;DR: In this article, the authors applied linear elastic theory to estimate the earthquake-induced soil pressures on a wall forming part of the structure of a power station founded on rock, and found that elasticity theory gave greater forces and moments than would be obtained by using the Mononobe-Okabe method.
Abstract: This paper describes the application of linear elastic theory to estimate the earthquake-induced soil pressures on a wall forming part of the structure of a power station founded on rock.
Analyses showed that the Mononobe-Okabe assumptions would not be applicable for this relatively rigid wall structure and it was found that elasticity theory gave greater forces and moments than would be obtained by using the Mononobe-Okabe method. The extent to which deformations of the structure and its foundations influence the wall pressures was investigated. It was found that even for this relatively rigid structure and foundation, the displacements resulting from the inertia of the wall structure can produce a significant increase in the total forces acting on the wall.
28 citations
TL;DR: In this article, the analysis of plane and axisymmetric problems with concrete-like material behavior where the weak tensile strength and crushing and cracking behavior are considered is dealt with, where the concrete is treated as a linear elastic, plastic, strain-hardening, and fracture material.
Abstract: This paper deals with the analysis of plane and axisymmetric problems with concrete-like material behavior where the weak tensile strength and crushing and cracking behavior are considered. The concrete is treated as a linear elastic, plastic, strain-hardening, and fracture material. The stress-strain relations are first stated in matrix form for the general case of three-dimensional stress states and then reduced to the special cases of plane stress, plane strain, and axisymmetric stress conditions, suitable for use in finite element analysis. The numerical analysis is obtained within the framework of the finite element method and a step- by-step integration procedure. The method is applied to the punch-indentation problems of concrete and metal blocks under plane strain conditions. Comparisons between analysis and experiment are made.
24 citations
TL;DR: In this article, a testing program to ascertain stress-strain, strain-rate, time, yield, and/or failure behavior of two adhesives (Metlbond 1113 and 1113-2) in bulk form using uniaxial tensile constant strain rate, creep, and relaxation tests is described.
Abstract: Studies on two adhesives (Metlbond 1113 and 1113-2) identified as having important applications in the bonding of composite materials are presented. A testing program to ascertain stress-strain, strain-rate, time, yield, and/or failure behavior of these materials in bulk form using uniaxial tensile constant strain-rate, creep, and relaxation tests is described. The stress-strain behavior of each material is shown to be significantly rate dependent. A rate dependent stress whitening (crazing) phenomenon occurs prior to either yield or fracture. A region of linear elasticity, a region of viscoelasticity, and the onset of yielding are identified in the stress-strain behavior. The linear elastic limit and the yield point are shown to be rate dependent and agree well with an empirical equation proposed by Ludwik. A creep to failure phenomenon is shown to exist and is correlated with a delayed yield equation proposed by Crochet. Analytical predictions based on a modified Bingham model are shown to agree well with experimental stress-strain strain-rate data. Analytical predictions based on a modified Ramberg-Osgood equation are also shown for comparison purposes.
22 citations
TL;DR: In this article, a hypothetical constitutive relation for EHL oil films is proposed which combines a linear elastic response with a non-linear shear thinning viscous response for small strains.
18 citations
01 Mar 1975
TL;DR: In this article, a class of mixed boundary-value problems for a linear elastic material subject to the internal constraint of inextensibility in a given direction is formulated and a uniqueness theorem is established.
Abstract: A class of mixed boundary-value problems is formulated for a linear elastic material subject to the internal constraint of inextensibility in a given direction. Due to the constraint, the usual prescription of boundary data has to be modified. A uniqueness theorem is established. For the particular cases of homogeneous isotropic and transversely isotropic materials, this theorem provides necessary and sufficient conditions for uniqueness of solution to the mixed problems posed.
TL;DR: In this article, a mew method is presented for the analysis of suspended structures in the presence of combined geometric and physical nonlinearities, and the solution can be attained by minimizing, under sign constraints only, an energy function of nodal displacements and "corrective" or "inelastic" strains.
Abstract: A mew method is presented for the analysis of suspended structures in the presence of combined geometric and physical nonlinearities. Physical nonlinearity is due to loosening of cable numbers or their tensile yielding, or both, and is described by piecewise linear laws. It is shown that the solution can be attained by minimizing, under sign constraints only, an energy function of nodal displacements and “corrective” or “inelastic” strains (i.e., strains additional to the linear elastic ones). Thus a new generalization of the potential energy principle is achieved. The proposed numerical procedure consists of a fairly efficient algorithm in nonlinear programming. The irreversible nature of possible plastic deformations can be easily allowed for, from one loading step to another, by adjusting the piecewise linear cable law.
TL;DR: In this paper, a simplified structural characterization of the human skull-brain system is formulated by using a linear elastic shell containing a linear core incorporating experimentally determined properties, and the model dynamic response for axisymmetric translational impact loading is sought by obtaining the shell-core static, free vibration, and time function solution.
Abstract: A simplified structural characterization of the human skull-brain system is formulated. The model is represented by a linear elastic shell containing a linear elastic core incorporating experimentally determined properties. The model dynamic response for axisymmetric translational impact loading is sought by obtaining the shell-core static, free vibration, and time function solution. Structural validation of the human head model is demonstrated by comparing the model static stiffness coefficient, fundamental eigenfrequency, radial dynamic displacement, brain pressure, and shear response with corresponding experimental results.
TL;DR: In this article, the uniqueness and existence of solutions are established for the traction problem for general domains. But the important case of prescribed tractions at each boundary point gives a non-standard potential problem involving linking of boundary values at opposite ends of chords parallel to the axis of material symmetry.
Abstract: L. W. MORLAND*(Received 8 July, 1974)(Revised 10 August, 1974)AbstractA plane strain or plane stress configuration of an inextensible transverselyisotropic linear elastic solid with the axis of symmetry in the plane, leads to aharmonic lateral displacement field in stretched coordinates. Various displace-ment and mixed displacement-traction boundary conditions yield standardboundary-value problems of potential theory for which uniqueness andexistence of solutions are well established. However, the important case ofprescribed tractions at each boundary point gives a non-standard potentialproblem involving linking of boundary values at opposite ends of chordsparallel to the axis of material symmetry. Uniqueness and existence ofsolutions, within arbitrary rigid motions, are now established for the tractionproblem for general domains.
TL;DR: In this paper, the static stresses in earth embankments were calculated by linear and nonlinear finite element procedures, and the results from both methods were in close agreement, except within the narrow cores of zone dams.
Abstract: A nonlinear incremental loading finite element analysis is the best currently available method for calculating the static stresses in earth embankments, and is virtually the only available method for calculating static deformations. However, if deformations are not required, the static stresses may be calculated by a simpler gravity-turn-on linear elastic finite element analysis. Examples are presented of four dams each with several different loading conditions, in which stresses were calculated by linear and by nonlinear finite element procedures. The results from both methods were in close agreement. The calculated stresses were virtually independent of the Young's modulus parameters, except within the narrow cores of zone dams. The horizontal shear and normal stresses were strongly influenced by the selected values of poisson's ratio. However, if consistent values were used, the linear and the nonlinear analyses gave almost identical results. Recognition of the reliability of linear methods can lead to substantial savings. /ASCE/
TL;DR: In this article, the influence of an external stress acting parallel to the crack front on the plastic zone size is calculated and demonstrated by examples using the v. Mises criterion and the Sneddon equations.
TL;DR: In this paper, the interaction energy contours and force-displacement relation for rigid drift of the dislocation were computed for silver and cadmium solute atoms in a zinc matrix and compared with those obtained by the isotropic treatment.
Abstract: The interaction due to the size misfit between a substitutional solute atom and an edge dislocation is treated, employing anisotropic linear elasticity. The interaction energy contours and force—displacement relation for rigid glide of the dislocation are computed for silver and cadmium solute atoms in a zinc matrix and compared with those obtained by the isotropic treatment. The results for cadmium are rather similar to those of the isotropic computation. However, for silver there are large differences in the shapes and magnitudes of the interaction energy and force relation between the isotropic and anisotropic treatments.
TL;DR: In this paper, the response of a reinforced concrete wall-beam framework to static lateral loading is examined, where the structure is first idealized as a plane frame composed of linear elastic material.
Abstract: The response of a staggered reinforced concrete wall-beam framework to static lateral loading is examined. The structure is first idealized as a plane frame composed of linear elastic material. The behavior of a single wall-beam element is than analyzed. Effective moments of inertia for the wall I-section, the lintel I-section, and the floor slabs are calculated. The influence of local deformations at the lintel to web junction are investigated. These results are combined to determine the properties of an equivalent uniform beam member. The axial stiffness of a column modified by the presence of a wall is also considered. The response of a wall-beam structure is described and static test results are presented. Good agreement between calculated and measured deflections was found.
TL;DR: In this article, the authors present a tractable formulation of a stress-strain relationship for engineering soils that incorporates a concise statement of nonlinearity, inelasticity, and stress dependency.
Abstract: During the past decade, increasing interest has been shown in the development of a satisfactory formulation of a stress-strain relationship for engineering soils that incorporates a concise statement of nonlinearity, inelasticity, and stress dependency. Conventional approaches have tended to assume a constant modulus of elasticity and Poisson.s ratio and although it was acknowledged that it was unwise to place too much reliance on linear elastic theory, few others tractable solutions have been available.
Journal Article•
TL;DR: In this paper, the tangent modulus at a given stress level is shown to be the slope of the axial stress-axial strain curve at the stress level and the value of Poisson's ratio is evaluated by use of theoretical considerations and a simple graphical construction.
Abstract: A method is suggested to determine piecewise linear, stress-dependent relationships for the modulus and Poisson's ratio of soils. The method is based on linear elasticity and conditions associated with conventional triaxial tests at different states of stress. The tangent modulus at a given stress level is shown to be the slope of the axial stress-axial strain curve at the stress level, and the value of Poisson's ratio is evaluated by use of theoretical considerations and a simple graphical construction. This method of interpretation is applied to experimental data from two natural soils used in an actual full-scale field installation of buried concrete pipe, and the results are shown to be in reasonable agreement with those determined by more sophisticated analyses and more extensive experimental measurements. It is also demonstrated that other analytical methods for interpreting these test data may yield significantly different values for the mechanical properties of soils, and this must be taken into account when such results are incorporated into mathematical models for the response of soil-structure systems.
01 Jun 1975
TL;DR: In this article, the diaphragm stepped thrust pad (l) operating at low rotor surface speeds is simulated on the PDP-6 digital computer according to a numerical procedure involving the solution of the Reynolds equation and the elasticity equation respectively by the finite element method and the finite difference method.
Abstract: The diaphragm stepped thrust pad (l)† operating at low rotor surface speeds is simulated on the PDP-6 digital computer according to a numerical procedure involving the solution of the Reynolds equation and the elasticity equation respectively by the finite element method and the finite difference method. The computed results compare very well with the experimental results. Multiple regression analysis is then applied to the computed data of the simulated pad to obtain design relationships between parameters affecting the performance of the pad.
TL;DR: In this article, the problem of the optimal (from the weight standpoint) longitudinal variation of the modulus of elasticity is considered with reference to a member in axial compression when the limit state is reached as a result of loss of stability.
Abstract: The problem of the optimal (from the weight standpoint) law of longitudinal variation of the modulus of elasticity is considered with reference to a member in axial compression when the limit state is reached as a result of loss of stability. Constraints are imposed on the modulus of elasticity. The problem is solved with the aid of the apparatus of the generalized maximum principle.
TL;DR: In this article, three sets of data for the deformation of crystalline poly-propylene in uniaxial tensile tests, in equal biaxia creep and in planar shear are examined.
Abstract: Three sets of data for the deformation of crystalline poly-propylene in uniaxial tensile tests and uniaxial creep, in equal biaxial creep and in planar shear are examined. The methods of analysis used in the original papers, all based on variants of linear elasticity, are critically evaluated, and a constitutive equation based on the theory of a simple fluid with memory is proposed.
TL;DR: The shape of a wire confined to the surface of a sphere with its ends tangent to two given circles of lattitude and otherwise free, in its most relaxed state, is found by solving a variational problem to be the solution of a fourth order nonlinear two point boundary value problem as mentioned in this paper.
TL;DR: In this article, a numerical method is described for the dynamic stress analysis of certain linear elastic fracture problems, based on the dynamic relaxation technique, may be used to derive an important quantity known in the theory of continuum fracture mechanics as the energy release rate G. The relevant of this quantity is discussed with regard to the prediction of crack paths in for example a rock breakage process.
TL;DR: In this paper, different sector elements for two-dimensional problems for linear elastic analysis are developed and their utility for different types of problems is discussed and a considerable accuracy of the results is obtained with a relatively small number of elements.
TL;DR: In this paper, the method of conformal transformation is used to consider a class of crack problems in linear isotropic elasticity, such as the straight line crack and the circular arc crack.
Abstract: In this paper the method of conformal transformation is used to consider a class of crack problem in linear isotropic elasticity. The conformal transformation used is the Joukowski transformation so that in general the crack profile is an aerofoil. This is a useful profile to consider since as well as providing information about the general aerofoil crack it includes as special cases the circular arc crack, the straight line crack and, as will be shown, can also be used to obtain similar results to those obtained by Bowie (1956) for the crack originating at the boundary of a circular hole. Thus by varying the parameters in the one elastic solution it is possible to obtain the elastic solution for a number of crack profiles. An examination of the stresses at the crack tip is made since this gives useful information about the direction of crack extension and also gives information necessary for the application of the fracture criterion which is derived in section 6. Some particular crack profiles are considered in detail in section 7.