By J. Tinsley Oden
A latest method of mathematical modeling, that includes special functions from the sphere of mechanics
An advent to Mathematical Modeling: A path in Mechanics is designed to survey the mathematical types that shape the rules of recent technology and accommodates examples that illustrate how the main winning versions come up from easy ideas in glossy and classical mathematical physics. Written via a global authority on mathematical concept and computational mechanics, the booklet offers an account of continuum mechanics, electromagnetic box concept, quantum mechanics, and statistical mechanics for readers with different backgrounds in engineering, desktop technology, arithmetic, and physics.
The writer streamlines a complete knowing of the subject in 3 truly prepared sections:
Nonlinear Continuum Mechanics introduces kinematics in addition to strength and pressure in deformable our bodies; mass and momentum; stability of linear and angular momentum; conservation of strength; and constitutive equations
Electromagnetic box idea and Quantum Mechanics features a short account of electromagnetic wave thought and Maxwell's equations in addition to an introductory account of quantum mechanics with similar themes together with ab initio equipment and Spin and Pauli's principles
Statistical Mechanics offers an creation to statistical mechanics of platforms in thermodynamic equilibrium in addition to continuum mechanics, quantum mechanics, and molecular dynamics
Each a part of the e-book concludes with workout units that permit readers to check their realizing of the awarded fabric. Key theorems and basic equations are highlighted all through, and an in depth bibliography outlines assets for additional study.
Extensively class-tested to make sure an available presentation, An creation to Mathematical Modeling is a wonderful booklet for classes on introductory mathematical modeling and statistical mechanics on the upper-undergraduate and graduate degrees. The ebook additionally serves as a useful reference for pros operating within the components of modeling and simulation, physics, and computational engineering.
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Extra info for An Introduction to Mathematical Modeling: A Course in Mechanics
E. the dependent variables are regarded as functions of spatial position x (and t). We now refer the fields to the reference configuration (see Fig. 10) P(X) = detF(X) T(x) F(X)T = T(x) Cof F(X) (by the Piola transformation), where the dependence on t has been suppressed. The tensor P(X) is called the First Piola—Kirchhoff Stress Tensor. Note that P is not symmetric; however, PFT = FPT since T is symmetric. 11) vP + f0)(X) dv 1 = Thus, the equations of motion (linear and angular momentum) referred to the reference configuration are Div P(X) + f0(X) = po(X)ü(X), P(X) FT(X) = F(X) PT(X).
6). Let T = T(x) denote a tensor field defined on G and T(x) n(x) Here the flux of T across 3G, n(x) being a unit normal to 3G. is fixed so t is held constant and not displayed. Corresponding to T, a tensor field T0 = T0(X) is defined on C0 that associates the 16 CHAPTER 1 KINEMATICS OF DEFORMABLE BODIES flux T0(X) no(X) through 3G0, no(X) being the unit normal to 3G0. 26) This relationship between T0 and T is called the Piola transformation. 1 (Piola Transformation) The above correspondence holds if T0(X) = detF(X) T(x) F(X)_T = T(x) CofF(X).
An Introduction to Mathematical Modeling A Course in Me hank s. First Edition By J Tinsley Oden © 2011 John Wiley & Sons, Inc Published 2011 by John Wiley & Sons. Inc 29 30 CHAPTER 3. 1: External contact forces (bottom left) and internal contact forces (bottom right). 1) Contact Forces Contact forces are also called surface forces because the contact of one body with another or with its surroundings must take place on a material surface. Contact forces fall into two categories (see Fig. 1): • External contact forces representing the contact of the exterior boundary surface of the body with the environment outside the body, and • Internal contact forces representing the contact of arbitrary parts of the body that touch one another on parts of internal surfaces they share on their common boundary.