Conservation and Constitutive Equations in Curvilinear Coordinates

2020 ◽  
pp. 225-233
Author(s):  
Gianpietro Elvio Cossali ◽  
Simona Tonini
Keyword(s):  
Constitutive Equations ◽  
Curvilinear Coordinates

Prediction of Tread Pattern Wear by an Explicit Finite Element Model3

2007 ◽  
Vol 35 (4) ◽  
pp. 276-299 ◽  
Author(s):  
J. C. Cho ◽  
B. C. Jung
Keyword(s):  
Finite Element ◽  
Wear Rate ◽  
Tangential Stress ◽  
Rate Equation ◽  
Constitutive Equations ◽  
Slip Velocity ◽  
Test Results ◽  
Explicit Finite Element ◽  
Driving Condition ◽  
Driving Conditions

Abstract Tread pattern wear is predicted by using an explicit finite element model (FEM) and compared with the indoor drum test results under a set of actual driving conditions. One pattern is used to determine the wear rate equation, which is composed of slip velocity and tangential stress under a single driving condition. Two other patterns with the same size (225/45ZR17) and profile are used to be simulated and compared with the indoor wear test results under the actual driving conditions. As a study on the rubber wear rate equation, trial wear rates are assumed by several constitutive equations and each trial wear rate is integrated along time to yield the total accumulated wear under a selected single cornering condition. The trial constitutive equations are defined by independently varying each exponent of slip velocity and tangential stress. The integrated results are compared with the indoor test results, and the best matching constitutive equation for wear is selected for the following wear simulation of two other patterns under actual driving conditions. Tens of thousands of driving conditions of a tire are categorized into a small number of simplified conditions by a suggested simplification procedure which considers the driving condition frequency and weighting function. Both of these simplified conditions and the original actual conditions are tested on the indoor drum test machines. The two results can be regarded to be in good agreement if the deviation that exists in the data is mainly due to the difference in the test velocity. Therefore, the simplification procedure is justified. By applying the selected wear rate equation and the simplified driving conditions to the explicit FEM simulation, the simulated wear results for the two patterns show good match with the actual indoor wear results.

Constitutive Equations of Soils: 4. Macro-Rheology

1978 ◽  
Vol 18 (3) ◽  
pp. 85-95
Author(s):  
Hideo Sekiguchi
Keyword(s):  
Constitutive Equations

Certain problems with the application of stochastic diffusion processes for description of chemical engineering phenomena; Diffusion equations in curvilinear coordinates

1991 ◽  
Vol 56 (3) ◽  
pp. 602-618
Author(s):  
Vladimír Kudrna
Keyword(s):  
Heat Transfer ◽  
Mass Transport ◽  
Probability Density ◽  
Chemical Engineering ◽  
Diffusion Processes ◽  
Curvilinear Coordinates ◽  
Diffusion Equations ◽  
Parabolic Partial Differential Equations ◽  
Spherical Coordinates ◽  
Transformation Procedure

Parabolic partial differential equations used in chemical engineering for the description of mass transport and heat transfer and analogous relationship derived in stochastic processes theory are given. A standard transformation procedure is applied, allowing these relations to be generally written in curvilinear coordinates and particular expressions for cylindrical and spherical coordinates to be derived. The relation between the probability density for the position of a discernible particle and the concentration of a set of such particles is discussed.

scholarly journals Shaping Microstructure and Mechanical Properties of High-Carbon Bainitic Steel in Hot-Rolling and Long-Term Low-Temperature Annealing

Materials ◽  
2021 ◽  
Vol 14 (2) ◽  
pp. 384
Author(s):  
Tomasz Dembiczak ◽  
Marcin Knapiński
Keyword(s):  
Mathematical Models ◽  
Hot Rolling ◽  
Constitutive Equations ◽  
Rolling Process ◽  
Bainitic Steel ◽  
Compression Tests ◽  
High Carbon ◽  
Bainite Steel ◽  
Dilatometric Studies

Based on the research results, coefficients in constitutive equations, describing the kinetics of dynamic, meta-dynamic, and static recrystallization in high-carbon bainitic steel during hot deformation were determined. The developed mathematical model takes into account the dependence of the changing kinetics in the structural size of the preliminary austenite grains, the value of strain, strain rate, temperature, and time. Physical simulations were carried out on rectangular specimens. Compression tests with a flat state of deformation were carried out using a Gleeble 3800. Based on dilatometric studies, coefficients were determined in constitutive equations, describing the grain growth of the austenite of high-carbon bainite steel under isothermal annealing conditions. The aim of the research was to verify the developed mathematical models in semi-industrial conditions during the hot-rolling process of high-carbon bainite steel. Analysis of the semi-industrial studies of the hot-rolling and long-term annealing process confirmed the correctness of the predicted mathematical models describing the microstructure evolution.

Rate Dependent Constitutive Equations of Cyclic Softening

1985 ◽  
Vol 40 (7) ◽  
pp. 653-665
Author(s):  
J. S. Mshana ◽  
A. S. Krausz
Keyword(s):  
Stress Relaxation ◽  
Low Temperature ◽  
Constitutive Equations ◽  
Strain Softening ◽  
Structural Characteristics ◽  
High Stress ◽  
Cyclic Strain ◽  
Cyclic Stress ◽  
Cyclic Softening ◽  
Stress Softening

Constitutive equations of cyclic strain and stress softening for materials with low internal stress levels are derived from the rate theory. The study shows that over the high stress and low temperature range where the description of plastic flow in cyclic softening can be approximated with activation over a single energy barrier, cyclic strain softening is well related to stress relaxation process while cyclic stress softening is related to creep process. The material structural characteristics for cyclic strain softening, cyclic stress softening and stress relaxation are identical. Subsequently, it is shown that cyclic stress and strain softening within the high stress and low temperature range can be evaluated from the constitutive equations using the material structural characteristics measured from a simple stress relaxation test.

Local material symmetry group for first- and second-order strain gradient fluids

2021 ◽  
pp. 108128652110216
Author(s):  
Victor A. Eremeyev
Keyword(s):  
Symmetry Group ◽  
Strain Gradient ◽  
Constitutive Equations ◽  
Strain Energy ◽  
Mass Density ◽  
Material Model ◽  
Second Order ◽  
Material Symmetry ◽  
Local Material ◽  
Current Mass

Using an unified approach based on the local material symmetry group introduced for general first- and second-order strain gradient elastic media, we analyze the constitutive equations of strain gradient fluids. For the strain gradient medium there exists a strain energy density dependent on first- and higher-order gradients of placement vector, whereas for fluids a strain energy depends on a current mass density and its gradients. Both models found applications to modeling of materials with complex inner structure such as beam-lattice metamaterials and fluids at small scales. The local material symmetry group is formed through such transformations of a reference placement which cannot be experimentally detected within the considered material model. We show that considering maximal symmetry group, i.e. material with strain energy that is independent of the choice of a reference placement, one comes to the constitutive equations of gradient fluids introduced independently on general strain gradient continua.

XVI.—On Triple Systems of Surfaces and Non-Orthogonal Curvilinear Coordinates

1927 ◽  
Vol 46 ◽  
pp. 194-205 ◽  
Author(s):  
C. E. Weatherburn
Keyword(s):  
Curvilinear Coordinates ◽  
Triple Systems ◽  
Orthogonal Curvilinear Coordinates ◽  
Orthogonal Systems ◽  
Considerable Detail

The properties of “triply orthogonal” systems of surfaces have been examined by various writers and in considerable detail; but those of triple systems generally have not hitherto received the same attention. It is the purpose of this paper to discuss non-orthogonal systems, and to investigate formulæ in terms of the “oblique” curvilinear coordinates u, v, w which such a system determines.

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