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WO2009116048A2 - Procédé pour la modélisation d’élément à point de cosserat (cpe) de matériaux élastiques non linéaires - Google Patents

Procédé pour la modélisation d’élément à point de cosserat (cpe) de matériaux élastiques non linéaires Download PDF

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Publication number
WO2009116048A2
WO2009116048A2 PCT/IL2009/000315 IL2009000315W WO2009116048A2 WO 2009116048 A2 WO2009116048 A2 WO 2009116048A2 IL 2009000315 W IL2009000315 W IL 2009000315W WO 2009116048 A2 WO2009116048 A2 WO 2009116048A2
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cpe
bending
deformations
constitutive
modeling
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WO2009116048A3 (fr
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Mahmood Jabareen
Miles B. Rubin
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Technion Research and Development Foundation Ltd
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Technion Research and Development Foundation Ltd
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    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G06F30/23—Design optimisation, verification or simulation using finite element methods [FEM] or finite difference methods [FDM]

Definitions

  • CPE COSSERAT POINT ELEMENT
  • the present invention relates to a generalized form for the strain energy of inhomogeneous deformations, and more particularly to a 3-D brick Cosserat Point Element (CPE) which includes full coupling of bending and torsional modes of deformation, such that the resulting constitutive coefficients ensure that the strain energy for inhomogeneous deformations remains a positive definite function of the inhomogeneous strain measures for all reference element shapes.
  • CPE Cosserat Point Element
  • the Cosserat Point Element is a finite element technology for the numerical solution of large deformation problems of elastic materials. Since the CPE can be introduced into general purpose finite element codes, it can be used to solve the whole host of problems considered in industry to design elastic structures. In particular, the CPE can be used for the problem of large elastic deformations of tires, which is particularly challenging, since the solutions based on other elements typically are corrupted by unphysical hourglass instabilities.
  • Fig. 1 is a prior art sketch of a general brick CPE showing the numbering of the nodes 110 and the surfaces 120.
  • the kinematics of the CPE are characterized by eight element director vectors 130 and the kinetics propose eight balance laws of director momentum to determine the dynamic response of the element.
  • the locations of the nodes in the current deformed configuration are characterized by eight nodal director vectors and the element directors are related to the nodal directors by standard tri-linear shape functions.
  • the CPE theory considers the element as a structure and introduces a strain energy function which characterizes the response of the structure. Also, the nodal forces are related to derivatives of the strain energy function through algebraic relations in a similar manner to the relationship of the stress to derivatives of the strain energy function in the full three- dimensional theory of hyperelastic materials.
  • Jabareen and Rubin (2007c) developed analytical expressions for constitutive coefficients in an improved CPE for 3-D deformations by generalizing the quadratic form of the strain energy function for inhomogeneous deformations to include additional coupling of the inhomogeneous strains active in bending modes. Functional forms for the additional coefficients were determined by considering four bending solutions for special shaped reference elements which can be represented as parallelepipeds with two right angles. Jabareen and Rubin (2007b) also developed improved coefficients for torsion by considering an exact torsion-like solution of the linear elastic equations for an isotropic elastic material.
  • Jabareen M, Rubin MB (2007a) Hyperelasticity and physical shear buckling of a block predicted by the Cosserat point element compared with inelasticity and hourglassing predicted by other element formulations. Computational Mechanics, 40: 447-459. Jabareen M, Rubin MB (2007b) Modified torsion coefficients for a 3-D brick Cosserat point element. Computational Mechanics 41: 517-525.
  • Jabareen M, Rubin MB (2007c) An improved 3-D Cosserat brick element for irregular shaped elements. Computational Mechanics 40: 979-1004. Jabareen M, Rubin MB (2008) A Cosserat point element (CPE) for nearly planar problems
  • Simo JC Rifai MS (1990) A class of mixed assumed strain methods and the method of incompatible modes. Mt. J. Numer. Meth. Engng. 29: 1595-1638. Simo JC, Armero F (1992) Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes. Int. J. Numer. Meth. Engng. 33: 1413-1449. Simo JC, Armero F, Taylor RL (1993) Improved versions of assumed enhanced strain tri- linear elements for 3D finite deformation problems. Comp. Meth. Appl. Mech. Engrg.
  • CPE 3-D brick Cosserat Point Element
  • a method for Cosserat Point Element (CPE) modeling of nonlinear elastic materials utilizing a strain energy function characterizing the response of an eight node 3-D brick element to homogeneous and inhomogeneous deformations, which includes full coupling of bending and torsional modes of deformation.
  • the method includes determining the constitutive coefficients which satisfy bending and torsional modes of deformation for three specific reference element geometries and combining these solutions to obtain constitutive equations for general reference element shapes which ensure that the strain energy remains a positive definite function for all inhomogeneous deformations of the CPE.
  • the constitutive coefficients characterize a generalized CPE model the response of a 3-D brick element for finite elastic deformations which is free of hourglass instabilities and locking for nearly incompressible materials, thereby providing a robust modeling tool.
  • the present invention provides a generalized form for the strain energy of inhomogeneous deformations, developed for a 3-D brick CPE, which includes full coupling of bending and torsional modes of deformation.
  • the constitutive coefficients which depend on the reference geometry of the element, are determined by solving eighteen bending problems and six torsion problems on special elements that are parallelepipeds with two right angles. The resulting constitutive coefficients ensure that the strain energy for inhomogeneous deformations remains a positive definite function of the inhomogeneous strain measures for all reference element shapes.
  • a number of exemplary problems are considered which show that the generalized CPE produces results as accurate as enhanced strain and incompatible elements for thin structures and is free of hourglass instabilities typically predicted by these enhanced elements in regions experiencing combined high compression with bending.
  • Section 2 of the summary presents the basic equations of the CPE.
  • Section 3 of the summary describes the procedure to use eighteen bending solutions and six torsions solutions to determine the functional forms of the constitutive coefficients in the strain energy function for inhomogeneous deformations.
  • Section 4 of the detailed description considers a number of illustrated examples that demonstrate the response of the generalized CPE. Section 5 presents conclusions.
  • the three-dimensional position vector X associated with material points in the reference configuration of the element can be expressed in the form
  • region P 0 associated with the reference configuration is mapped to the region
  • the element directors and nodal directors are related using a tri-linear form like (2.2) for the position vector x of material points in the present configuration. This causes the displacement field to be continuous across element boundaries.
  • a strain energy function for the structure and do not depend on the point-wise validity of this approximate expression for x*.
  • the specific (per unit mass) strain energy ⁇ of the CPE can be expressed in terms of the specific strain energy ⁇ * of the three-dimensional material and the specific strain energy ⁇ associated with inhomogeneous deformations, such that where the kinematic quantities are defined by
  • 2007b can be expressed in the form
  • Each of these element shapes is a parallelepiped with two right angles. Now, from (Nadler and Rubin, 2003) it follows that for these element shapes the reference geometry is characterized by and the position vector X H in (2.2) and the Cartesian coordinates X j ' can be expressed in the forms
  • Analytical expressions for B 1 - can be developed by matching the solutions (3.1) and (3.2) for each of the element shapes (3.3). Then, the resulting coefficients are combined in a manner that ensures that B-- is a positive definite tensor. Specifically, with reference to the element shape El in (3.3a) consider six bending solutions associated with specifications of the orientation of e- relative to D 1
  • Fig. 1 is a prior art sketch of a general brick CPE showing the numbering of the nodes and the surfaces;
  • Fig. 2 is a schematic illustration of the cross-section of a parallelepiped element El, constructed according to the principles of the present invention
  • Fig. 3 is a schematic illustration of a shear load on a thin cantilever beam, where the irregular element mesh is based on the distorted center cross-section, constructed according to the principles of the present invention
  • Figs. 4a-4d show the response of a shear load on a thin cantilever beam for small deformations, constructed in accordance with the principles pf the present invention
  • Fig. 5 shows a sketch of a thin slanted cantilever beam with dimensions (4.4) and with the slanting angle ⁇ , constructed in accordance with the principles of the present invention
  • Fig. 7 shows a shear load on a thin twisted cantilever beam where the element mesh is based on the distorted center cross-section, and is constructed in accordance with the principles of the present invention
  • Figs. 8a - 8e are graphs for shear loading of a thin twisted cantilever beam with a pre- twist ⁇ and the mesh ⁇ 20nx5n ⁇ n ⁇ (small deformations), plotted in accordance with the principles of the present invention
  • Fig. 11 is a schematic illustration of a point load on the corner of a thin partially clamped rhombic plate, constructed according to the principles of the present invention
  • Figs. 12a - 12c are graphs of the response to a point load on the corner of a thin partially clamped rhombic plate for small deformations, plotted according to the principles of the present invention
  • Fig. 13 is a sketch of a point load on the center of a thin fully clamped square plate with an irregular element mesh, constructed according to the principles of the present invention
  • Figs 14a and b are graphs of the response to a point load on the center of a thin fully clamped square plate with small deformations, plotted according to the principles of the present invention
  • Figs. 16a - 16d are graphs of the response to a point load on a partially clamped rhombic plate for large deformations, plotted according to the principles of the present invention
  • Fig. 17 shows a sketch of one eighth of a thin circular cylindrical shell that is subjected to a pair of opposing point loads P, constructed according to the principles of the present invention
  • Figs. 18a and 18b are graphs of the force P versus radial displacement u r of the point A under the load for different mesh refinements, plotted according to the principles of the present inventions showing that ⁇ G, F ⁇ tend to converge to the same solutions;
  • Fig. 20 is a schematic illustration of plane strain indentation of a rigid plate into a nearly incompressible block showing the boundary conditions and definition of element irregularity for this problem, constructed according to the principles of the present invention
  • Figs. 21a and 21b are graphs of the response of plane strain indentation of a rigid plate into a nearly incompressible block, plotted according the principles of the present invention
  • Figs. 23a - 23e represent plane strain indentation of a rigid plate into a nearly incompressible block showing graphs of nonlinear load curves using the regular mesh ⁇ 8nx4nxl ⁇ for different values of n, plotted according to the principles of the present invention
  • Fig. 25 is a schematic representation of indentation of a rigid plate into a block showing the boundary conditions and definition of element irregularity, where symmetry conditions are used so that only one fourth of the block is meshed, and constructed according to the principles of the present invention
  • Figs. 27a - 27d are graphic representations of the response to indentation of a rigid plate into a nearly incompressible block, plotted according to the principles of the present invention.
  • Fig. 2 a schematic illustration of the cross-section of a parallelepiped element El 200, constructed according to the principles of the present invention, it can be seen that the torsion expressions (3.16a) and (3.16b) are similar to those in (Nadler and Rubin, 2003), since the cross-section normal to D 1 210 has lengths and the cross-section normal to D 2 220 has lengths .
  • the value of ⁇ can be eliminated in the resulting equations of equilibrium (3.13) and the value of ⁇ can be eliminated from each of the equations of equilibrium associated with the torsion solutions.
  • the values (3.16) are used in the resulting torsion equations.
  • strain energy function ⁇ in (2.11) will be positive definite provided that the coefficient matrix By is positive definite.
  • Fig. 3 is a schematic illustration of the shear load on a thin cantilever beam, where the irregular element mesh is based on the distorted center cross-section, constructed according to the principles of the present invention.
  • the lateral surfaces are traction free.
  • the mesh ⁇ 20nxnxn ⁇ is defined by distorting the middle cross-section in its reference configuration (using the parameters a 1 ⁇ a 2 , a 3 , a 4 shown in Fig. 3), with 1On elements on each side of this cross-section and n elements in each of the e 2 320 and e 3 330 directions. Two cases of element distortion are considered
  • Figs. 4a - 4d show the results for a shear load on a thin cantilever beam for small deformations, plotted in accordance with the principles pf the present invention.
  • Figs. 4a and 4c show errors in the displacement of the point A in the e 2 direction versus variation in the distortion parameter a/H 410 and
  • Figs. 4b and 4d show the errors versus n for the mesh ⁇ 20nxnxn ⁇ defined for two cases of element distortion.
  • This figure shows that the predictions of (G) and (I) are identical and are slightly more accurate than those of (F) for large values of irregularity.
  • the results for Case II shown in Figs. 4c and 4d indicate that (G) is again slightly more accurate than (F), and that they both converge to the same solution.
  • Fig. 5 shows a sketch of a thin slanted cantilever beam 500 with dimensions (4.4) and with the slanting angle ⁇ 510, constructed in accordance with the principles of the present invention.
  • the boundary conditions are the same as those for the previous example except that the shear load P 520 is applied in the e 3 530 direction to emphasize differences between the predictions of (G) and (I).
  • the mesh is taken to be ⁇ 20nxnxn ⁇ with 2On elements in the axial direction of the beam. All of the elements have parallelogram cross-sections in the C 1 -C 2 plane, with sides parallel to the ends of the beam.
  • Fig. 7 shows a shear load on a thin twisted cantilever beam where the element mesh is based on the distorted center cross-section, and is plotted in accordance with the principles of the present invention.
  • the problem of shear loading of a thin twisted beam provides a severe test of the accuracy of an element formulation since the elements have irregular shapes and the response couples torsion and bending modes of deformation.
  • the twisted beam In its unstressed reference configuration the twisted beam has length L 711 and a rectangular cross-section with height H 712 and width W 713 given by
  • each of the cross-sections is twisted by the angle ⁇ such that the position of a material point X * in the reference configuration is given by
  • Figs. 8a - 8e are graphs of the results for shear loading of a thin twisted cantilever beam with a pre-twist ⁇ and the mesh ⁇ 20nx5nxn ⁇ (small deformations), plotted in accordance with the principles of the present invention. The influence of element irregularity is shown in
  • the direction e ⁇ of the load is held constant during loading.
  • the buckling process is triggered by the small pre-twist
  • the value L corresponds to the actual length of each edge and the value of P is given by
  • the mesh used for the plate is defined by ⁇ IOnxiOnxn ⁇ with n elements through the thickness.
  • Figs. 12a - 12c are graphs of the response to a point load on the corner of a thin partially clamped rhombic plate for small deformations, plotted according to the principles of the present invention.
  • Fig. 13 is a sketch of one quarter of a thin fully clamped square plate that is loaded by a point force P 1310 at its center, constructed according to the principles of the present invention. Dimensions are given in (4.8). Only one quarter of the plate is modeled and the value of force P 1310 given by (4.16) corresponds to one quarter of the load that would be applied to the center of the entire plate. Irregular elements are specified by moving the center point of the quarter section to the position characterized by the lengths ⁇ a 1? a 2 ⁇ , as shown in Fig. 13, and defined by the two cases
  • the quarter section of the plate is meshed by ⁇ IOnxiOnxn ⁇ with each subsection being meshed by ⁇ 5nx5nxn ⁇ and with n elements through the thickness.
  • Figs. 14a and b are graphs of the point load on the center of a thin fully clamped square plate with small deformations, plotted according to the principles of the present invention. Errors in the displacement of the point A in the e 3 direction versus the distortion parameters
  • the plate is fully clamped on two edges and the other edges and major surfaces are traction free.
  • the dimensions are given by (4.8) as shown in Fig. 11, with L now being the length of the plate's edge, and the point force P is
  • Figs. 16a - 16d are graphs of the response to a point load on a partially clamped rhombic plate for large deformations, plotted according to the principles of the present invention.
  • Fig. 17 shows a sketch of one eighth of a thin circular cylindrical shell that is subjected to a pair of opposing point loads P 1710, constructed according to the principles of the present invention.
  • the entire shell has length 2L, middle surface radius R 1720, and thickness H
  • Fig. 19 shows the deformed shape of one eighth of the circular shell 1910 predicted by
  • Fig. 20 is a schematic illustration of plane strain indentation of a rigid plate into a nearly incompressible block showing the boundary conditions and definition of element irregularity for this problem, constructed according to the principles of the present invention.
  • the block has length 2L, height L and depth W. Material points on its sides and bottom remain in contact with a rigid container and are allowed to slide freely.
  • the top surface of the block is loaded by a rigid plate (AB) of length L which makes perfect contact with the block so that material points in contact with the rigid plate move only vertically. The remaining half of the block's top surface is traction free.
  • Irregular meshes are defined by dividing the block into four subsections, with the central node moving to the position characterized by the lengths ⁇ a ⁇ , a 2 ⁇ , as shown in Fig. 20, and as defined by two cases
  • the entire block is meshed by ⁇ 8nx4nxl ⁇ with 4n elements in the C 1 direction 2010 and 2n elements in the e 2 direction 2020 in each of the subsections.
  • the point C 2030 is located on the free top surface at a distance 0.25 L from the corner B 2040 of the rigid plate. Also, the material is considered to be nearly incompressible.
  • Fig. 21 shows the convergence of this error predicted by ⁇ G, QlPO, HO9 ⁇ . This error is plotted relative to n for the mesh ⁇ 8n ⁇ 4n ⁇ l ⁇ in Fig. 21a and is plotted relative to the degrees of freedom (DOF, calculated for plane strain response) in Fig. 21b. From Fig. 21a it is not clear if (QlPO) exhibits a locking behavior by converging to a value different from (G) or whether the convergence rate is very slow.
  • Figs. 23a — 23e represent plane strain indentation of a rigid plate into a nearly incompressible block showing graphs of nonlinear load curves using the regular mesh ⁇ 8n ⁇ 4n ⁇ l ⁇ for different values of n, plotted according to the principles of the present invention.
  • Fig. 23a shows nonlinear load curves of (G) using the regular mesh ⁇ 8n ⁇ 4n ⁇ l ⁇ for different values of n.
  • Fig. 23b shows nonlinear load curves of (QlPO) using the regular mesh ⁇ 8nx4nxl ⁇ for different values of n.
  • Figs. 23a — 23e represent plane strain indentation of a rigid plate into a nearly incompressible block showing graphs of nonlinear load curves using the regular mesh ⁇ 8n ⁇ 4n ⁇ l ⁇ for different values of n, plotted according to the principles of the present invention.
  • Fig. 23a shows nonlinear load curves of (G) using the regular mesh ⁇ 8
  • u ⁇ -0.15 (2410 and 2420), -0.20 (2430 and 2440) and -0.25 (2450 and 2460), respectively.
  • the flexibility of (G) allows the elements near the plate's corner to roll around the corner more easily than allowed by (QlPO).
  • Fig. 25 is a schematic representation of indentation of a rigid plate into a block showing the boundary conditions and definition of element irregularity, where symmetry conditions are used so that only one fourth of the block is meshed, and constructed according to the principles of the present invention.
  • Fig. 25 shows a sketch of one fourth of a nearly incompressible block that has total length 2L, height L and depth 2L with
  • the remaining portion of the block's top surface is traction free. Irregular elements are generated by moving the nodes of the center plane by the displacements ⁇ a l5 a ⁇ , a 3 , a 4 ⁇ as shown in Fig. 25 with
  • the mesh for the one fourth region is specified by ⁇ 4nx4n ⁇ 4n ⁇ with 2n elements below and above the distorted center surface and with ⁇ nxn ⁇ elements under the rigid plate. Moreover, the vertical reference locations of material points on these distorted surfaces are described by a bi-linear form of the coordinates (X ⁇ , X ⁇ ). Furthermore, the point E 2530 is located at a distance L/4 from the edge of the rigid plate.
  • Figs. 27a - 27d 2710, 2720, 2730 and 2740, respectively, are graphic representations of the response to indentation of a rigid plate into a nearly incompressible block, plotted according to the principles of the present invention.
  • Fig. 27 shows nonlinear load curves using the regular mesh ⁇ 4nx4n ⁇ 4n ⁇ for different values of n. From these figures it can be seen that the results predicted by ⁇ G, Q1P0 ⁇ are similar. In particular, the unphysical stiffness exhibited by (QlPO) in the previous plane strain problem is not detected in this three-dimensional problem.
  • the response of a structure to a specified load should be insensitive to irregularities in the element shapes used to mesh the structure.
  • the results here indicate that the main features of this desired response for general shaped elements can be obtained by properly modeling pure bending and simple torsion of parallelepipeds with two right angles.
  • insensitivity to element irregularity can best be exhibited by focusing attention on a thin structure that is loaded so that inhomogeneous deformations (like bending and torsion) dominate homogeneous deformations.
  • plots like Fig. 4c for (I) clearly emphasize undesirable sensitivity to element irregularity.
  • the nodal forces in the generalized CPE are determined by algebraic expressions in terms of derivatives of a strain energy function and no integration is needed over the element region.
  • a number of example problems also see Jabareen and Rubin, 2007a,c) have been considered which show that the generalized CPE is as accurate as elements based on enhanced strain and incompatible modes and is as robust as elements based on full integration.
  • the plane strain example of indentation of a rigid plate into a nearly incompressible block showed that the flexibility exhibited by the CPE is physical and that the mixed QlPO element predicts unphysical stiffness.
  • the generalized CPE can be used to model 3-D bodies, thin shells and rods and nearly incompressible materials.
  • the generalized CPE is free of hourglass instabilities that are observed in other element formulations in regions experiencing combined high compression with bending. Consequently, the generalized CPE is truly a robust user friendly element that can be used with confidence to model problems in nonlinear elasticity.
  • a tire deformation detection system comprises a deformation detection structure, a transmitter, and receiver.
  • US Pat. No 4,475,384, by Christie, Tire Sidewall Deformation Detection Techniques describes an improved tire manufacturing system, including means for testing the extent of sidewall deformation of a tire. The tire is rotated and measured for lateral run-out of both sidewalls. The resulting data is used to increase the accuracy by which unacceptable sidewall bulges and valleys are detected.
  • US Pat. No 7,412,879 by Serra, et al., Method for Monitoring Tyre Deformations and
  • Monitoring System Therefor, teaches a method for monitoring deformations in a tire of a vehicle wheel, the wheel including the tire and a rim, the method including: providing the wheel with at least two magnetic-field emitters mutually disposed so as to yield a composite magnetic field and monitoring variations in the magnetic field and correlating the variation with tire deformations.
  • Slip Angle is the angle between a rolling wheel's actual direction of travel and the direction towards which it is pointing (i.e., the angle of the vector sum of wheel translational velocity v and sideslip velocity u). This slip angle results in a force perpendicular to the wheel's direction of travel ⁇ the cornering force.
  • Tire load sensitivity describes the behavior of tires under load.
  • the maximum horizontal force developed should be proportional to the vertical load on the tire.
  • Cornering force is the sideways force produced by a vehicle tire during cornering.
  • Load Transfer is the imaginary "shifting" of weight around a motor vehicle during acceleration and deceleration.
  • Weight Transfer is the redistribution of weight supported by each tire during acceleration or deceleration. Weight transfer is a crucial concept in understanding vehicle dynamics. Slip ratio When a vehicle is being driven along a road in a straight line its wheels rotate at virtually identical speeds. The vehicle's body also travels along the road at this same speed.
  • Scrub Radius is the lateral distance measured in front or rear view between the center of the tire contact patch and the intersection of the steering axis with the ground. Scrub radius and kingpin inclination determine the moment arm about the steering axis for longitudinal (braking and acceleration) forces acting at the tire contact patch.
  • Axes of measurement Tire forces are divided into three axes: radial, lateral, and tangential.
  • the radial axis runs from the tire center toward the tread, and is the vertical axis running from the roadway through the tire center toward the vehicle. This axis supports the vehicle's weight.
  • the lateral axis runs sideways across the tread.
  • Radial force variation acts upward to support the vehicle, radial force variation describes the change in this force as the tire rotates under load. As the tire rotates and spring elements with different spring constants enter and exit the contact area, the force will change.
  • RFV can be shown as a complex waveform. This waveform can be expressed according to its harmonics by applying Fourier Transform (FT).
  • FT Fourier Transform
  • Lateral force variation acts side-to-side along the tire axle, lateral force variation describes the change in this force as the tire rotates under load.
  • Tangential force variation acts in the direction of travel, and describes the change in this force as the tire rotates under load.
  • Conicity is based on lateral force behavior. It is the characteristic that describes the tire's tendency to roll like a cone.
  • PIysteer is a parameter based on lateral force behavior. It is the characteristic that is usually described as the tire's tendency to "crab walk", or move sideways while maintaining a straight-line orientation.
  • Radial runout describes the deviation of the tire's roundness from a perfect circle.
  • RRO can be expressed as the peak-to-peak value as well as harmonic values.
  • LRO Lateral runout describes the deviation of the tire's sidewall from a perfect plane.
  • LRO can be expressed as the peak-to-peak value as well as harmonic values.
  • Tread Shear Displacement is the amount the tread particles are displaced from their at rest center. By examining the picture under "slip,” one can notice that the tread shear displacement is represented by the contact pitch, which is offset.
  • Total Shear Displacement is the combined amount the tread, sidewall/carcass and belt are displaced from their at rest center line.
  • Tread Shear Angle the angle of tread particle displacement within the contact patch.
  • Sidewall Shear Angle is the angle of sidewall, carcass and belt (radial tire) displacement from their at rest center line.
  • Total Shear Angle the combined tread and sidewall/carcass angle of shear.
  • Trailing Edge Slip Percentage is the percentage of tread particles exceeding the coefficient of friction at the trailing edge of the contact patch in relation to the contact patch's length.
  • Tread Shear Angle the angle in which, during tire deformation, the tread particles are moved in the different of the lateral forces.
  • Tread Particles do not act independently of each other but rather push and pull one against the next much like the fibers of a cleaning brash. In the case of tire tread depth, a shorter fiber of equal strength will product a higher force for an equal amount of shear.

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Abstract

La présente invention concerne un procédé pour la modélisation d’élément à point de Cosserat (CPE) de matériaux élastiques non linéaires, à l’aide d’une fonction d’énergie de déformation caractérisant la réponse d’un élément de brique en 3D à huit nœuds à des déformations homogènes et hétérogènes, qui comprend le couplage complet de modes de déformation en courbure et en torsion. Le procédé comprend la détermination des coefficients constitutifs qui satisfont aux modes de déformation en courbure et en torsion pour trois géométries élémentaires de référence spécifiques et la combinaison de ces solutions pour obtenir des équations constitutives pour des formes élémentaires de référence générales qui garantissent que l’énergie de déformation reste une fonction définie positive pour toutes les déformations hétérogènes du CPE. Les coefficients constitutifs caractérisent un modèle CPE généralisé qui prédit la réponse d’un élément de briques en 3D pour des déformations élastiques finies qui est dépourvue d’instabilités en sablier et de blocage pour des matériaux pratiquement incompressibles, donnant ainsi un puissant outil de modélisation.
PCT/IL2009/000315 2008-03-20 2009-03-19 Procédé pour la modélisation d’élément à point de cosserat (cpe) de matériaux élastiques non linéaires Ceased WO2009116048A2 (fr)

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Cited By (4)

* Cited by examiner, † Cited by third party
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WO2019029130A1 (fr) * 2017-08-10 2019-02-14 北京汽车集团有限公司 Accessoire de véhicule en matériau composite à armure tridimensionnelle et procédé de fabrication de celui-ci
CN113420370A (zh) * 2021-06-02 2021-09-21 中国航空工业集团公司沈阳飞机设计研究所 一种高度静不定结构的强度设计载荷获取方法
CN117216862A (zh) * 2023-11-09 2023-12-12 湖南大学 一种基于纤维梁单元的三维盾构隧道动力分析模型

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WO2013032773A1 (fr) * 2011-09-01 2013-03-07 Mcmoran Oil & Gas, Llc Méthodologie mise au point par des ingénieurs pour la vérification et la validation modélisées d'équipements de contrôle de puits de pétrole et de gaz à haute température et à haute pression en ultra grande profondeur
US8688418B2 (en) 2011-09-01 2014-04-01 Mcmoran Oil & Gas, Llc Engineered methodology for design verification and validation of ultra-deep high pressure high temperature oil and gas well control equipment
WO2019029130A1 (fr) * 2017-08-10 2019-02-14 北京汽车集团有限公司 Accessoire de véhicule en matériau composite à armure tridimensionnelle et procédé de fabrication de celui-ci
CN113420370A (zh) * 2021-06-02 2021-09-21 中国航空工业集团公司沈阳飞机设计研究所 一种高度静不定结构的强度设计载荷获取方法
CN113420370B (zh) * 2021-06-02 2024-03-19 中国航空工业集团公司沈阳飞机设计研究所 一种高度静不定结构的强度设计载荷获取方法
CN117216862A (zh) * 2023-11-09 2023-12-12 湖南大学 一种基于纤维梁单元的三维盾构隧道动力分析模型
CN117216862B (zh) * 2023-11-09 2024-02-02 湖南大学 一种基于纤维梁单元的三维盾构隧道动力分析模型

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