j n reddy introduction to continuum mechanics
Introduction to Continuum Mechanics 3E pptx
... Introduction to Continuum Mechanics This page intentionally left blank Introduction to Continuum Mechanics Third Edition W MICHAEL LAI Professor of Mechanical Engineering and Orthopaedic ... indicating a summation with the index running through the integers 1,2, , n This convention is known as Einstein's summation convention Using the convention, Eq (2A1.1) shortens to We also note that It ... tensors, then Tp.-1-Sp are components of a third order tensor To prove this rule, we note that since Tljk=QmiQnjQrkTmnr and S;jk=QmiQnjQrkSmnr we have, *ijk+Sijk = QmiQnjQrk*mnr+QmiQnjQrkTmnr...
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chorin a., marsden j.e. mathematical introduction to fluid mechanics
... proofreading of the first edition and to many other readers for supplying both corrections and support, in particular V Dannon, H Johnston, J Larsen, M Olufsen, and T Ratiu and G Rublein These corrections, ... ξ ∂y ψ = J( ξ, ψ), the Jacobian of ξ and ψ Thus, the flow is stationary (time independent) if and only if ξ and ψ are functionally dependent (If functional dependence holds at one instant it will ... definite functional form to σ For more information, see J Jeans [1867] An Introduction to the Kinetic Theory of Gases, Cambridge Univ Press 32 The Equations of Motion As before, Newton’s second...
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Introduction to Continuum Mechanics II doc
... Introduction to Continuum Mechanics This page intentionally left blank Introduction to Continuum Mechanics Third Edition W MICHAEL LAI Professor of Mechanical Engineering and Orthopaedic ... indicating a summation with the index running through the integers 1,2, , n This convention is known as Einstein's summation convention Using the convention, Eq (2A1.1) shortens to We also note that It ... tensors, then Tp.-1-Sp are components of a third order tensor To prove this rule, we note that since Tljk=QmiQnjQrkTmnr and S;jk=QmiQnjQrkSmnr we have, *ijk+Sijk = QmiQnjQrk*mnr+QmiQnjQrkTmnr...
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Introduction to Continuum Mechanics 3 Episode 1 ppsx
... Introduction to Continuum Mechanics This page intentionally left blank Introduction to Continuum Mechanics Third Edition W MICHAEL LAI Professor of Mechanical Engineering and Orthopaedic ... indicating a summation with the index running through the integers 1,2, , n This convention is known as Einstein's summation convention Using the convention, Eq (2A1.1) shortens to We also note that It ... Cartesian Components of a Tensor Defining Tensors by Transformation Laws Symmetric and Antisymmetric Tensors The Dual Vector of an Antisymmetric Tensor Eigenvalues and Eigenvectors of a Tensor Principal...
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Introduction to Continuum Mechanics 3 Episode 2 doc
... eigenvalue A Then, by definition, Tnj = An^ and Tn2 = An2 so that for any a, and ft, T(an1+/ 3n2 )=aTn1+/TTn2=A(ani+/ ?n2 ) That is ctn1-» -j8 n2 is also an eigenvector with the same eigenvalue A In ... equation has roots A! and A2=A3=A (Aj distinct from A) Let nj^ be the eigenvector corresponding to Aj, then nj is perpendicular to any eigenvector of A Now, corresponding to A, the equations 44 Tensors ... then Tp.-1-Sp are components of a third order tensor To prove this rule, we note that since Tljk=QmiQnjQrkTmnr and S;jk=QmiQnjQrkSmnr we have, *ijk+Sijk = QmiQnjQrk*mnr+QmiQnjQrkTmnr ~ QmiQn}Qrk(^mnr+^nmr)...
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Introduction to Continuum Mechanics 3 Episode 3 ppsx
... was in the direction n, is given by n • En In particular, if the element was in the ej direction in the reference state, then n = ej, and EH = ej • Eej so that EH is the unit elongation for an element ... (2D3.24c) 3 Kinematics of a Continuum The branch of mechanics in which materials are treated as continuous is known as continuum mechanics Thus, in this theory, one speaks of an infinitesimal volume ... T, R, and S are related by T - RS Tensors R and S have the same eigenvector n and corresponding eigenvalues rx and jj_ Find an eigenvalue and the corresponding eigenvector of T 2B32 If n is a...
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Introduction to Continuum Mechanics 3 Episode 4 docx
... material element in the direction of n, its rate of extension (i.e., rate of change of length per unit length ) is given by Dnn(no sum on n) The rate of extension is also known as stretching In particular ... Conditions for Infinitesimal Strain Components When any three displacement functions MI, u^, and u$ are given, one can always determine dUj the six strain components in any region where the partial ... displacementstrain equations We now state the following theorem: If EifiX\JtiJQ are continuous functions having continuous second partial derivatives in a simply connected region, then the necessary and sufficient...
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Introduction to Continuum Mechanics 3 Episode 5 docx
... unit elongation in the direction 2ej + 2e2 + 63? (b) What is the change of angle between two perpendicular lines (in the undeformed state) emanating from the point and in the directions of 2ej ... that convention, for example, TI\ and T2^ are tangential components of the stress vector on the plane whose normal is 62 etc These differences in meaning regarding the nondiagonal elements of T ... tne stationary value of 7j VI If I *-ftlf/ \Sttr * But the dni,dn2 and dn$ can not vary independently Indeed, taking the total derivative of Eq (4.6.6), i.e., nj+nl+wi = we obtain If we let and...
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Introduction to Continuum Mechanics 3 Episode 6 docx
... on planes defined by the unit vectors m and n and pass through the point P Show that if k is a unit vector that determines a plane that contains !„, and tn, then t,,, is perpendicular to m and ... tensor, E is the infinitesimal strain tensor, with T (0) = If in addition, the function is to be linear, then we have, in component form The above nine equations can be written compactly as Since ... equation can be written and in Cartesian component form Equations of Motion - Principle of Linear Momentum 189 These are the equations that must be satisfied for any continuum in motion, whether...
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Introduction to Continuum Mechanics 3 Episode 7 potx
... principal directions of stress and strain coincide (b) Find a relation between the principal values of stress and strain Solution, (a) Let nj be an eigenvector of the strain tensor E (i.e., Enj ... polarized normal to the plane of incidence, no longitudinal component occurs Also, when an incident longitudinal wave is reflected, in addition to the regularly reflected longitudinal wave, there ... is often not known, only the resultant force is known The question naturally arises under what conditions can an elasticity solution such as the one we just obtained for simple extension be applicable...
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Introduction to Continuum Mechanics 3 Episode 8 pps
... Internal and External Pressure 5.18 Thick-walled Circular Cylinder under internal and External Pressure Consider a circular cylinder subjected to the action of an internal pressure p/ and an ...
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Introduction to Continuum Mechanics 3 Episode 9 pptx
... nine independent coefficients can be inverted to give a symmetric compliance matrix with also nine independent constants The compliance matrix is The meanings of the constants in the compliance ... the transverse train in the x2 direction when stressed in the jtj direction or transverse strain in the jcj direction when stressed in the x2 direction (i.e., Poisson's ratio in the plane of isotropy, ... engineering constants: Also, 5.31 Engineering Constants for a Monoclinic Elastic Solid For a rnonoclinic elastic solid, the symmetric stiffness matrix with thirteen independent coefficients can be inverted...
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Introduction to Continuum Mechanics 3 Episode 10 pptx
... following equations of motion in terms of velocity components i.e., Or, in invariant form: Newtonian Viscous Fluid 361 These are known as the Navier-Stokcs Equations of motion for incompressible Newtonian ... polymeric solutions, require a more general model (Non-Newtonian Fluids) for an adequate description Non-Newtonian fluid models will be discussed in Chapter 6.1 Fluids Based on the notion of fluidity ... dp/dz is a constant Let then 376 Plane-Poiseuille Flow Thus, and Since v must be bounded in the flow region, the integration constant b must be zero Now, the nonslip condition on the cylindrical wall...
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Introduction to Continuum Mechanics 3 Episode 11 pptx
... vector £ as The tensor 2W is known as the vorticity tensor It can be easily seen that in indicial notation, the Cartesian components of? are and in invariant notation, In cylindrical coordinates ... 414 One-Dimensional Flow of a Compressible Fluid We now study the flow in a converging nozzle and the flow in a converging-diverging nozzle, using one-dimensional assumptions (1) Flow in a Converging ... be consistent with the state of stress corresponding to the state of rest and also to be consistent with the definition that/? is not to depend explicitly on any kinematic quantities when in motion,...
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Introduction to Continuum Mechanics 3 Episode 12 potx
... (7.1.1) and (7.1.2) to vj and v2 and adding, we have In indicial notation, Eq (7.2 la) becomes and in invariant notation, The following generalization not only appears natural, but can indeed be ... across a unit area, then the principle states: the minus sign in the last term is due to the convention that n is an outward unit normal vector and therefore -q -n represents inflow Again, using the ... first a one-dimensional problem in which the motion of a continuum, in Cartesian coordinates, is given by and the density field is given by The integral 434 integral Formulation of General Principles...
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Introduction to Continuum Mechanics 3 Episode 13 pptx
... function and the two normal stress functions can be obtained to be Non-Newtonian Fluids 501 A special nonlinear viscoelastic fluid defined by Eq (8.15.3) with a memory function dependent on the ... Similarly, one can obtain Non-Newtonian Fluids 483 and Equations (iii) to (v) are equivalent to the following equations: As already noted in the previous section, the matrix being obtained using bases ... frame, transforms to A* in the starred frame in accordance with the relation Non-Newtonian Fluids 495 then, the tensor A is said to be objective, or frame indifferent (i.e., independent of observers)...
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Introduction to Continuum Mechanics 3 Episode 14 pdf
... constant For a Newtonian fluid, such as water, the simple shearing flow gives Non-Newtonian Fluids 523 For a non-Newtonian fluid, such as a polymeric solution, for small k, the viscometric functions ... Evaluating the above equation at r—t and noting that and we obtain immediately (B) Oldroyd lower converted derivative Let us consider the tensor Non-Newtonian Fluids §09 Again, as in (A), and is an ... (damping coefficient rj0) and a Kelvin-Voigt solid (damping coefficient rj and spring constant G, see the previous problem) connected in series Also, obtain the relaxation function 8.5 A linear...
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Introduction to Continuum Mechanics 3 Episode 15 pptx
... Leigh, D.C., Nonlinear Continuum Mechanics, McGraw-Hill, New York, 1978 11 Malvern, L.E., Introduction to the Mechanics of a Continuous Medium, Prentice Hall, Englewood Cliffs, New Jersey, 1969 ... Fluid Mechanics, Wiley & Sons, New York, 1977 Coleman, B.D, Markowitz, H and W Noll, Viscometric Flows of Non-Newtonian Fluids, Springer-Verlag, New York, 1966 Eringen, A.C., Mechanics ofContinua, ... coordinates, 236 spherical coordinates, 236 Navier-Stokes equations cylindrical coordinates, 364 incompressible fluid, 360 spherical coordinates, 365 Newtonian fluid, 355 Non-Newtonian fluid, 462 Normal...
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solutions for an introduction to the finite element method (3rd edition), by j. n. reddy
... pertinent places in this manual Additional examples and problems can be found in the following books of the author: J N Reddy and M L Rasmussen, Advanced Engineering Analysis, John Wiley, New ... and 7) J N Reddy, Theory and Analysis of Elastic Plates, Taylor and Francis, Philadelphia, 1997 J N Reddy, Energy Principles and Variational Methods in Applied Mechanics, Second Edition, John ... = aji (i, j = 1, 2) and f are given functions of position (x, y) in a twodimensional domain Ω, and u0 and t0 are known functions on portions Γ1 and Γ2 of the boundary Γ: Γ1 + Γ2 = Γ Solution:...
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Introduction to fluid mechanics - P1
... designed and its Fig 1.2 Relief of ancient Egyptian ship The beginning of fluid mechanics Fig 1.3 Ancient Greek ship depicted on old vase inclination or supply pressure had to be adjusted to overcome ... separation region Leonardo was the first to find the least resistive ‘streamline’ shape 4 History of fluid mechanics Leonardo da Vinci (1452-1519 ) An all-round genius born in Italy His unceasing ... and incomparable power of imagination are apparent in numerous sketches and astonishing design charts of implements, precise human anatomical charts and flow charts of fluids He drew streamlines...
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