MOHR S CIRCLES FOR NON-SYMMETRIC STRESSES AND COUPLE STRESSES

Size: px
Start display at page:

Download "MOHR S CIRCLES FOR NON-SYMMETRIC STRESSES AND COUPLE STRESSES"

Transcription

1 CRNOGORSKA AKADEMIJA NAUKA I UMJETNOSTI GLASNIK ODJELJENJA PRIRODNIH NAUKA, 16, QERNOGORSKAYA AKADEMIYA NAUK I ISSKUSTV GLASNIK OTDELENIYA ESTESTVENNYH NAUK, 16, THE MONTENEGRIN ACADEMY OF SCIENCES AND ARTS PROCEEDINGS OF THE SECTION OF NATURAL SCIENCES, 16, 2005 UDK Vlado A. Lubarda MOHR S CIRCLES FOR NON-SYMMETRIC STRESSES AND COUPLE STRESSES A b s t r a c t Determination of the principal values of a non-symmetric stress tensor in plane strain problems of couple stress theory based on Mohr s circle construction is presented. It is shown that the antisymmetric component of the stress tensor affects the maximum shear stress, but not the maximum normal stress. The analysis is then extended to non-symmetric couple stresses under conditions of anti-plane strain. MOHROVI KRUGOVI ZA NESIMETRIČNE NAPONE I NAPONSKE SPREGOVE I z v o d U radu je prezentovana procedura za odredjivanje glavnih napona nesimetricnog tenzora napona u ravnom problemu deformacije na bazi Prof. dr V.A. Lubarda, The Montenegrin Academy of Sciences and Arts, Podgorica, SCG, and University of California, San Diego, CA , USA.

2 2 V.A. Lubarda konstrukcije Mohrovog kruga. Pokazano je da antisimetrična komponenta tenzora napona utiče na maksimalni smičući napon, ali ne i na maksimalni normalni napon. Analiza je proširena na slučaj nesimetricnih naponskih spregova u uslovima antiravne deformacije. 1. INTRODUCTION In a micropolar continuum the deformation is described by the displacement vector and an independent rotation vector (Eringen, 1968,1999; Nowacki, 1986). In the couple stress theory, the rotation vector ϕ i is not independent of the displacement vector u i but subject to the constraint ϕ i = 1 2 e ijk ω jk = 1 2 e ijk u k,j, ω ij = e ijk ϕ k, (1.1) as in classical continuum mechanics (Mindlin and Tiersten, 1962; Koiter, 1964). The skew-symmetric alternating tensor is e ijk, and ω ij are the rectangular components of the infinitesimal rotation tensor. The comma designates the partial differentiation with respect to Cartesian coordinates x i. The gradient of the rotation is a non-symmetric curvature tensor κ ij = ϕ j,i = e jkl ɛ ik,l. (1.2) Since ɛ ij is symmetric and e ijk is skew-symmetric, the curvature tensor in couple stress theory is a deviatoric tensor (κ kk = 0). A surface element ds transmits a force vector T i ds and a couple vector M i ds. The surface forces are in equilibrium with a non-symmetric Cauchy stress t ij, and the surface couples are in equilibrium with a non-symmetric couple stress m ij, such that T i = n j t ji, M i = n j m ji, (1.3) where n j are the components of the unit vector orthogonal to the surface element under consideration. In the absence of body forces and body couples, the differential equations of equilibrium are t ji,j = 0, m ji,j + e ijk t jk = 0. (1.4)

3 Mohr s Circles for Couple Stresses 3 By decomposing the stress tensor into its symmetric and antisymmetric part t ij = σ ij + τ ij, it readily follows from the moment equilibrium equation that τ ij = 1 2 e ijkm lk,l. (1.5) If the gradient of the couple stress vanishes at some point, the stress tensor is symmetric at that point. The normal stress in the plane orthogonal to the direction n is t n = σ n = σ ij n i n j, (1.6) because τ ij n i n j = 0 in view of the symmetry of n i n j. Thus, the principal stresses of the symmetric tensor σ ij are also the principal stresses of the nonsymmetric tensor t ij, although there are shear stresses in the principal planes of t ij due to antisymmetric shear stress components τ ij (Lubarda, 2003). The magnitude of this shear stress is τ 2 n = n i n j τ ik τ jk = τ τ τ 2 31 (n 1 τ 23 + n 2 τ 31 + n 3 τ 12 ) 2. (1.7) Similarly, the couple stress component m n = m ij n i n j is independent of the antisymmetric part of m ij. In the case if isotropic linear elasticity, σ ij = 2µ ɛ ij + λ ɛ kk δ ij, (1.8) m ij = 4α κ ij + 4β κ ji. (1.9) where µ, λ, α, and β are the Lamé-type constants of couple stress elasticity. In this case, since κ kk = 0, the couple stress is a deviatoric tensor (m kk = 0). 2. MOHR S CIRCLE FOR NON-SYMMETRIC STRESSES IN PLANE STRAIN For the plane strain problems, the displacement components are u 1 = u 1 (x 1, x 2 ), u 2 = u 2 (x 1, x 2 ), and u 3 = 0. The stress and couple

4 4 V.A. Lubarda t 22 m 23 t 21 t rr t 11 m 13 m 13 t 11 m 13 j t rj m r3 t 11 t 21 m 23 t 21 m 23 (a) t 22 t 22 (b) Figure 1: (a) A rectangular material element under conditions of plane strain; (b) an inclined plane at an angle ϕ supports the stresses t ρρ and t ρϕ, and the couple stress m ρ3. stress tensors are accordingly t 11 0 t = t 21 t t 33, m = 0 0 m m 23 m 31 m (2.1) Consider a material element with sides parallel to coordinates directions x 1 and x 2 (Fig. 1a). In an inclined plane whose normal ρ makes an angle ϕ with the direction x 1, the normal and shear stresses are t ρρ and t ρϕ, and the couple stress is m ρ3 (Fig. 1b). From the equilibrium conditions of the triangular element it readily follows that m ρ3 = m 13 cos ϕ + m 23 sin ϕ, (2.2) t ρρ = 1 2 (t 11 + t 22 ) (t 11 t 22 ) cos 2ϕ ( + t 21 ) sin 2ϕ, (2.3) t ρϕ = 1 2 ( t 21 ) ( + t 21 ) cos 2ϕ 1 2 (t 11 t 22 ) sin 2ϕ. (2.4) The planes with the maximum magnitude of t ρρ and t ρϕ are defined

5 Mohr s Circles for Couple Stresses 5 by tan 2ϕ 1 = + t 21, tan 2ϕ 2 = 2t 11, (2.5) 2t 11 + t 21 with the obvious connection ϕ 2 = ϕ 1 ± π/4. The corresponding extreme values of the stresses are t max ρρ = 1 2 (t 11 + t 22 ) ± 1 (t11 t 22 ) 2 + ( + t 21 ) 2, (2.6) 2 t max ρϕ = 1 2 ( t 21 ) ± 1 (t11 t 22 ) 2 + ( + t 21 ) 2. (2.7) 2 It is noted that t max ρρ depends only on the symmetric part of the stress tensor t ij. Indeed, if we decompose t into its symmetric and antisymmetric parts, σ = σ 11 σ 12 0 σ 12 σ σ 33, τ = where σ 11 = t 11, σ 22 = t 22, σ 33 = t 33, and we can write t max ρρ t max ρϕ 0 τ 12 0 τ , (2.8) σ 12 = 1 2 ( + t 21 ), τ 12 = 1 2 ( t 21 ), (2.9) = σ max ρρ = 1 2 (σ 11 + σ 22 ) ± 1 2 (σ11 σ 22 ) 2 + 4σ12 2, (2.10) = τ 12 ± σρϕ max, σρϕ max = 1 (σ11 σ 22 ) 2 + 4σ (2.11) 2 The physical interpretation of Eq. (2.11) is facilitated by observing that the shear stress on any inclined plane due to antisymmetric stress component τ 12 is also equal to τ 12. This follows from the force equilibrium condition for the triangular element shown in Fig. 2b. The moment equilibrium is ensured by the non-uniform field of couple stresses (not shown in Fig. 2). Equations (2.3) and (2.4) can be combined to give t ρρ 1 2 (t 11 + t 22 )] 2 + = 1 4 t ρϕ 1 2 ( t 21 ) ] 2 (t11 t 22 ) 2 + ( + t 21 ) 2], (2.12)

6 6 V.A. Lubarda j t t (a) (b) Figure 2: (a) A rectangular material element carrying the antisymmetric shear stress τ 12 ; (b) the corresponding shear stress at an arbitrarily inclined plane is also τ 12. which defines Mohr s circle for the non-symmetric stress components under conditions of plane strain (Fig. 3). The normal stress t ρρ = (t 11 + t 22 )/2 acts in the planes where t ρϕ attains its maximum or minimum value, while the shear stress t ρϕ = ( t 21 )/2 acts in the planes where t ρρ has its maximum or minimum value. If = t 21, Eq. (2.12) defines the classical Mohr s circle for a symmetric stress tensor (Timoshenko and Goodier, 1970) Eigenvalue Analysis The extreme values of the stresses t ρρ and t ρϕ can also be determined by an eigenvalue analysis. If n = {n 1, n 2 } is the unit vector perpendicular to the plane which supports t ρρ and t ρϕ = ( t 21 )/2, we can write n j t ji = λ n i ( t 21 ) ˆn i, (i, j = 1, 2). (2.13) The unit vector orthogonal to n is ˆn = { n 2, n 1 }. The system of

7 Mohr s Circles for Couple Stresses 7 t max rj 0 -t ( t 22, 21) 1 -( t11 22) 2 + 2j 2 2j ( t12 - t ) 21 t max rr t 2j 1 t rr ( t 11, ) t min rj t rj Figure 3: Mohr s circle for non-symmetric stresses in plane strain. The center of the circle is at the point with the coordinates 1 2 (t 11 + t 22 ), ( t 21 )]. The radius of the circle is 1 2 (t 11 t 22 ) 2 +( +t 21 ) 2. The angles ϕ 1, ϕ 2, and ϕ 0 specify the planes of t max ρρ, t max ρϕ, and t ρϕ = 0. equations (2.13) has a nontrivial solution for (n 1, n 2 ) if λ is given be the right-hand side of Eq. (2.6). The corresponding planes are defined by tan ϕ 1 = t 11 t 22 + t 21 ± ( ) ] t11 t 2 1/ , (2.14) + t 21 in agreement with the first expression from Eq. (2.5). Similarly, for the plane which supports the stresses t ρϕ and t ρρ = (t 11 + t 22 )/2, we have n j t ji = λ ˆn i (t 11 + t 22 ) n i, (i, j = 1, 2). (2.15) The system of equations (2.15) has a nontrivial solution for (n 1, n 2 ) if λ is given by the right-hand side of Eq. (2.7). The corresponding

8 8 V.A. Lubarda planes are defined by tan ϕ 2 = + t 21 t 11 t 22 ± ( ) ] t12 + t 2 1/ , (2.16) t 11 t 22 in agreement with the second expression from Eq. (2.5). The planes with the vanishing t ρϕ (if they exist) are found by solving the eigenvalue problem which specifies λ = t ρρ = 1 2 (t 11 + t 22 ) ± 1 2 The corresponding planes are defined by n j t ji = λ n i, (2.17) (t11 t 22 ) t 21. (2.18) tan ϕ 0 = 1 2t 21 { (t 11 t 22 ) ± (t 11 t 22 ) t 21 }. (2.19) There is one such plane if (t 11 t 22 ) 2 = 4 t 21, and two such planes if (t 11 t 22 ) 2 > 4 t 21. The planes with the vanishing t ρρ (if they exist) are found by solving the eigenvalue problem which specifies λ = t ρϕ = 1 2 ( t 21 ) ± 1 2 The corresponding planes are defined by n j t ji = λ ˆn i, (2.20) (t12 + t 21 ) 2 4t 11 t 22. (2.21) tan ϕ 0 = 1 2t 22 { ( + t 21 ) ± ( + t 21 ) 2 4t 11 t 22 }, (2.22) provided that ( + t 21 ) 2 4t 11 t 22.

9 Mohr s Circles for Couple Stresses 9 3. MOHR S CIRCLE FOR COUPLE STRESSES IN ANTI-PLANE STRAIN For the anti-plane strain problems, the displacement components are u 1 = u 2 = 0, and u 3 = u 3 (x 1, x 2 ). The stress and couple stress tensors have the components 0 0 t 13 t = 0 0 t 23 t 31 t 32 0, m = m 11 m 12 0 m 21 m (3.1) A material element under conditions of anti-plane strain is shown in Fig. 4a. In an inclined plane whose normal ρ makes an angle ϕ with the longitudinal direction x 1, the shear stress is t ρ3 and the couple stresses are m ρρ and m ρϕ (Fig. 4b). From the equilibrium conditions of the triangular element it readily follows that t ρ3 = t 13 cos ϕ + t 23 sin ϕ, (3.2) m ρρ = 1 2 (m 11 + m 22 ) (m 11 m 22 ) cos 2ϕ (m 12 + m 21 ) sin 2ϕ, (3.3) m ρϕ = 1 2 (m 12 m 21 ) (m 12 + m 21 ) cos 2ϕ 1 2 (m 11 m 22 ) sin 2ϕ. (3.4) The planes with the maximum magnitude of m ρρ and m ρϕ are defined by tan 2ϕ 1 = m 12 + m 21 2m 11, tan 2ϕ 2 = 2m 11 m 12 + m 21. (3.5) The two angles are related by ϕ 2 = ϕ 1 ± π/4. extreme values of the couple stress components are The corresponding m max ρρ = 1 2 (m 11 + m 22 ) ± 1 (m11 m 22 ) 2 + (m 12 + m 21 ) 2, (3.6) 2

10 10 V.A. Lubarda m 22 m 21 m rr m 11 m 12 t 13 t 23 t 23 t 13 m 12 m 11 m 11 m 12 m rj 13 tt r3 23 j t m 21 m 21 m 22 m 22 (a) (b) Figure 4: (a) A rectangular material element under conditions of antiplane strain; (b) an inclined plane at an angle ϕ supports the shear stress t ρ3 and couple stresses m ρρ and m ρϕ. m max ρϕ = 1 2 (m 12 m 21 ) ± 1 (m11 m 22 ) 2 + (m 12 + m 21 ) 2. (3.7) 2 Again, it may be noted that m max ρρ does not depend on the antisymmetric part of the couple stress tensor, i.e., on the component (m 12 m 21 )/2. Equations (3.3) and (3.4) can be combined to give m ρρ (m 11 + m 22 )] + m ρϕ 1 ] 2 2 (m 12 m 21 ) = 1 4 (m11 m 22 ) 2 + (m 12 + m 21 ) 2]. (3.8) This defines Mohr s circle for the couple stresses in anti-plane strain (Fig. 5). The couple stress component m ρρ = (m 11 + m 22 )/2 acts in the planes where m ρϕ attains its maximum or minimum value, while m ρϕ = (m 12 m 21 )/2 acts in the planes where m ρρ has its maximum

11 Mohr s Circles for Couple Stresses 11 0,-m ( m22 21) m max rj 1-2 ( m11+ m ) 22 2j ( m12-m ) 21 m max rr 2j 1 2j 2 m rr ( m 11, m 12 ) m min rj m rj Figure 5: Mohr s circle for couple stresses in anti-plane strain. The center of the circle is at the point with the coordinates 1 2 (m 11 + m 22 ), (m 12 m 21 )]. The radius of the circle is 1 2 (m 11 m 22 ) 2 + (m 12 + m 21 ) 2. The angles ϕ 1, ϕ 2, and ϕ 0 specify the planes of m max ρρ, m max ρϕ, and m ρϕ = 0. or minimum value. The planes for which m ρϕ = 0 are defined by tan ϕ 0 = 1 2m 21 { (m 11 m 22 ) ± (m 11 m 22 ) 2 + 4m 12 m 21 }. (3.9) There is one such plane if (m 11 m 22 ) 2 = 4m 12 m 21, and two such planes if (m 11 m 22 ) 2 > 4m 12 m 21. The normal component of the couple stress in these planes is m ρρ = 1 2 (m 11 + m 22 ) ± 1 2 (m11 m 22 ) 2 + 4m 12 m 21. (3.10) The planes for which m ρρ = 0 are defined by

12 12 V.A. Lubarda m max rj,-m ( m22 21) 0 2j 2 2j 0 1 -( m12 m ) 2 - m max 21 rr 2j 1 m rr ( m 11, m 12 ) m min rj m rj Figure 6: Mohr s circle for couple stresses in anti-plane strain in the case when the couple stress is a deviatoric tensor. The center of the circle is along the m ρϕ axis at the distance 1 2 (m 12 m 21 ) from the origin. The radius of the circle is m max ρρ = m (m 12 + m 21 ) 2. tan ϕ 0 = 1 2m 22 { (m 12 + m 21 ) ± (m 12 + m 21 ) 2 4m 11 m 22 }, (3.11) provided that (m 12 + m 21 ) 2 4m 11 m 22. The non-vanishing couple stress in these planes is m ρϕ = 1 2 (m 12 m 21 ) ± 1 2 (m12 + m 21 ) 2 4m 11 m 22. (3.12) The extreme values of the couple stresses m ρρ and m ρϕ can also be determined by an eigenvalue analysis. The derivation is analogous to that presented in section 2. The resulting formulas can be obtained from Eqs. (2.13) (2.22) by replacing the stress symbol t with the couple stress symbol m. In the case when the couple stress is a deviatoric tensor (e.g., isotropic linear elasticity), the results simplify due to the

13 Mohr s Circles for Couple Stresses 13 condition m 11 + m 22 = 0. The corresponding Mohr s circle is shown in Fig. 6. Acknowledgment: Research support from the Montenegrin Academy of Sciences and Arts is kindly acknowledged. REFERENCES Eringen, A.C. (1968) Theory of micropolar elasticity. In Fracture An Advanced Treatise (Ed. H. Liebowitz), pp Academic Press, New York. Eringen, A.C. (1999) Microcontinuum Field Theories, Springer-Verlag, New York. Koiter, W.T. (1964) Couple-stresses in the theory of elasticity. Proc. Ned. Akad. Wet. (B) 67, Part I: 17 29, Part II: Lubarda, V.A. (2003) Circular inclusions in anti-plane strain couple stress elasticity. Int. J. Solids Struct. 40, Mindlin, R.D. and Tiersten, H.F. (1962) Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal. 11, Nowacki, W. (1986) Theory of Asymmetric Elasticity, (translated by H. Zorski). Pergamon Press, Oxford, and PWN - Polish Sci. Publ., Warszawa. Timoshenko, S. and Goodier, J.N. (1970) Theory of Elasticity. McGraw- Hill, New York.

PEAT SEISMOLOGY Lecture 2: Continuum mechanics

PEAT SEISMOLOGY Lecture 2: Continuum mechanics PEAT8002 - SEISMOLOGY Lecture 2: Continuum mechanics Nick Rawlinson Research School of Earth Sciences Australian National University Strain Strain is the formal description of the change in shape of a

More information

3D Elasticity Theory

3D Elasticity Theory 3D lasticity Theory Many structural analysis problems are analysed using the theory of elasticity in which Hooke s law is used to enforce proportionality between stress and strain at any deformation level.

More information

By drawing Mohr s circle, the stress transformation in 2-D can be done graphically. + σ x σ y. cos 2θ + τ xy sin 2θ, (1) sin 2θ + τ xy cos 2θ.

By drawing Mohr s circle, the stress transformation in 2-D can be done graphically. + σ x σ y. cos 2θ + τ xy sin 2θ, (1) sin 2θ + τ xy cos 2θ. Mohr s Circle By drawing Mohr s circle, the stress transformation in -D can be done graphically. σ = σ x + σ y τ = σ x σ y + σ x σ y cos θ + τ xy sin θ, 1 sin θ + τ xy cos θ. Note that the angle of rotation,

More information

Stress, Strain, Mohr s Circle

Stress, Strain, Mohr s Circle Stress, Strain, Mohr s Circle The fundamental quantities in solid mechanics are stresses and strains. In accordance with the continuum mechanics assumption, the molecular structure of materials is neglected

More information

Symmetry and Properties of Crystals (MSE638) Stress and Strain Tensor

Symmetry and Properties of Crystals (MSE638) Stress and Strain Tensor Symmetry and Properties of Crystals (MSE638) Stress and Strain Tensor Somnath Bhowmick Materials Science and Engineering, IIT Kanpur April 6, 2018 Tensile test and Hooke s Law Upto certain strain (0.75),

More information

Chapter 3 Stress, Strain, Virtual Power and Conservation Principles

Chapter 3 Stress, Strain, Virtual Power and Conservation Principles Chapter 3 Stress, Strain, irtual Power and Conservation Principles 1 Introduction Stress and strain are key concepts in the analytical characterization of the mechanical state of a solid body. While stress

More information

Mechanics PhD Preliminary Spring 2017

Mechanics PhD Preliminary Spring 2017 Mechanics PhD Preliminary Spring 2017 1. (10 points) Consider a body Ω that is assembled by gluing together two separate bodies along a flat interface. The normal vector to the interface is given by n

More information

16.20 Techniques of Structural Analysis and Design Spring Instructor: Raúl Radovitzky Aeronautics & Astronautics M.I.T

16.20 Techniques of Structural Analysis and Design Spring Instructor: Raúl Radovitzky Aeronautics & Astronautics M.I.T 16.20 Techniques of Structural Analysis and Design Spring 2013 Instructor: Raúl Radovitzky Aeronautics & Astronautics M.I.T February 15, 2013 2 Contents 1 Stress and equilibrium 5 1.1 Internal forces and

More information

Lecture 15 Strain and stress in beams

Lecture 15 Strain and stress in beams Spring, 2019 ME 323 Mechanics of Materials Lecture 15 Strain and stress in beams Reading assignment: 6.1 6.2 News: Instructor: Prof. Marcial Gonzalez Last modified: 1/6/19 9:42:38 PM Beam theory (@ ME

More information

Elements of Rock Mechanics

Elements of Rock Mechanics Elements of Rock Mechanics Stress and strain Creep Constitutive equation Hooke's law Empirical relations Effects of porosity and fluids Anelasticity and viscoelasticity Reading: Shearer, 3 Stress Consider

More information

Linearized theory of elasticity

Linearized theory of elasticity Linearized theory of elasticity Arie Verhoeven averhoev@win.tue.nl CASA Seminar, May 24, 2006 Seminar: Continuum mechanics 1 Stress and stress principles Bart Nowak March 8 2 Strain and deformation Mark

More information

3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1

3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 Math Problem a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 3 6 Solve the initial value problem u ( t) = Au( t) with u (0) =. 3 1 u 1 =, u 1 3 = b- True or false and why 1. if A is

More information

Mechanics of solids and fluids -Introduction to continuum mechanics

Mechanics of solids and fluids -Introduction to continuum mechanics Mechanics of solids and fluids -Introduction to continuum mechanics by Magnus Ekh August 12, 2016 Introduction to continuum mechanics 1 Tensors............................. 3 1.1 Index notation 1.2 Vectors

More information

Classical fracture and failure hypotheses

Classical fracture and failure hypotheses : Chapter 2 Classical fracture and failure hypotheses In this chapter, a brief outline on classical fracture and failure hypotheses for materials under static loading will be given. The word classical

More information

Engineering Sciences 241 Advanced Elasticity, Spring Distributed Thursday 8 February.

Engineering Sciences 241 Advanced Elasticity, Spring Distributed Thursday 8 February. Engineering Sciences 241 Advanced Elasticity, Spring 2001 J. R. Rice Homework Problems / Class Notes Mechanics of finite deformation (list of references at end) Distributed Thursday 8 February. Problems

More information

Elastic Wave Theory. LeRoy Dorman Room 436 Ritter Hall Tel: Based on notes by C. R. Bentley. Version 1.

Elastic Wave Theory. LeRoy Dorman Room 436 Ritter Hall Tel: Based on notes by C. R. Bentley. Version 1. Elastic Wave Theory LeRoy Dorman Room 436 Ritter Hall Tel: 4-2406 email: ldorman@ucsd.edu Based on notes by C. R. Bentley. Version 1.1, 20090323 1 Chapter 1 Tensors Elastic wave theory is concerned with

More information

1. Tensor of Rank 2 If Φ ij (x, y) satisfies: (a) having four components (9 for 3-D). (b) when the coordinate system is changed from x i to x i,

1. Tensor of Rank 2 If Φ ij (x, y) satisfies: (a) having four components (9 for 3-D). (b) when the coordinate system is changed from x i to x i, 1. Tensor of Rank 2 If Φ ij (x, y satisfies: (a having four components (9 for 3-D. Φ i j (x 1, x 2 = β i iβ j jφ ij (x 1, x 2. Example 1: ( 1 0 0 1 Φ i j = ( 1 0 0 1 To prove whether this is a tensor or

More information

Basic concepts to start Mechanics of Materials

Basic concepts to start Mechanics of Materials Basic concepts to start Mechanics of Materials Georges Cailletaud Centre des Matériaux Ecole des Mines de Paris/CNRS Notations Notations (maths) (1/2) A vector v (element of a vectorial space) can be seen

More information

Properties of the stress tensor

Properties of the stress tensor Appendix C Properties of the stress tensor Some of the basic properties of the stress tensor and traction vector are reviewed in the following. C.1 The traction vector Let us assume that the state of stress

More information

Tensor Transformations and the Maximum Shear Stress. (Draft 1, 1/28/07)

Tensor Transformations and the Maximum Shear Stress. (Draft 1, 1/28/07) Tensor Transformations and the Maximum Shear Stress (Draft 1, 1/28/07) Introduction The order of a tensor is the number of subscripts it has. For each subscript it is multiplied by a direction cosine array

More information

A short review of continuum mechanics

A short review of continuum mechanics A short review of continuum mechanics Professor Anette M. Karlsson, Department of Mechanical ngineering, UD September, 006 This is a short and arbitrary review of continuum mechanics. Most of this material

More information

MATH45061: SOLUTION SHEET 1 V

MATH45061: SOLUTION SHEET 1 V 1 MATH4561: SOLUTION SHEET 1 V 1.) a.) The faces of the cube remain aligned with the same coordinate planes. We assign Cartesian coordinates aligned with the original cube (x, y, z), where x, y, z 1. The

More information

CVEN 5161 Advanced Mechanics of Materials I

CVEN 5161 Advanced Mechanics of Materials I CVEN 5161 Advanced Mechanics of Materials I Instructor: Kaspar J. Willam Revised Version of Class Notes Fall 2003 Chapter 1 Preliminaries The mathematical tools behind stress and strain are housed in Linear

More information

Fundamentals of Linear Elasticity

Fundamentals of Linear Elasticity Fundamentals of Linear Elasticity Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research of the Polish Academy

More information

2 Basic Equations in Generalized Plane Strain

2 Basic Equations in Generalized Plane Strain Boundary integral equations for plane orthotropic bodies and exterior regions G. Szeidl and J. Dudra University of Miskolc, Department of Mechanics 3515 Miskolc-Egyetemváros, Hungary Abstract Assuming

More information

SEMM Mechanics PhD Preliminary Exam Spring Consider a two-dimensional rigid motion, whose displacement field is given by

SEMM Mechanics PhD Preliminary Exam Spring Consider a two-dimensional rigid motion, whose displacement field is given by SEMM Mechanics PhD Preliminary Exam Spring 2014 1. Consider a two-dimensional rigid motion, whose displacement field is given by u(x) = [cos(β)x 1 + sin(β)x 2 X 1 ]e 1 + [ sin(β)x 1 + cos(β)x 2 X 2 ]e

More information

1 Introduction. 1.1 Contents of book

1 Introduction. 1.1 Contents of book 1 Introduction 1.1 Contents of book An introduction to the use of anisotropic linear elasticity in determining the static elastic properties of defects in crystals is presented. The defects possess different

More information

Module #4. Fundamentals of strain The strain deviator Mohr s circle for strain READING LIST. DIETER: Ch. 2, Pages 38-46

Module #4. Fundamentals of strain The strain deviator Mohr s circle for strain READING LIST. DIETER: Ch. 2, Pages 38-46 HOMEWORK From Dieter 2-7 Module #4 Fundamentals of strain The strain deviator Mohr s circle for strain READING LIST DIETER: Ch. 2, Pages 38-46 Pages 11-12 in Hosford Ch. 6 in Ne Strain When a solid is

More information

Lecture 8. Stress Strain in Multi-dimension

Lecture 8. Stress Strain in Multi-dimension Lecture 8. Stress Strain in Multi-dimension Module. General Field Equations General Field Equations [] Equilibrium Equations in Elastic bodies xx x y z yx zx f x 0, etc [2] Kinematics xx u x x,etc. [3]

More information

Bulletin of the Transilvania University of Braşov Vol 10(59), No Series III: Mathematics, Informatics, Physics, 83-90

Bulletin of the Transilvania University of Braşov Vol 10(59), No Series III: Mathematics, Informatics, Physics, 83-90 Bulletin of the Transilvania University of Braşov Vol 10(59), No. 1-2017 Series III: Mathematics, Informatics, Physics, 83-90 GENERALIZED MICROPOLAR THERMOELASTICITY WITH FRACTIONAL ORDER STRAIN Adina

More information

Surface force on a volume element.

Surface force on a volume element. STRESS and STRAIN Reading: Section. of Stein and Wysession. In this section, we will see how Newton s second law and Generalized Hooke s law can be used to characterize the response of continuous medium

More information

Continuum mechanism: Stress and strain

Continuum mechanism: Stress and strain Continuum mechanics deals with the relation between forces (stress, σ) and deformation (strain, ε), or deformation rate (strain rate, ε). Solid materials, rigid, usually deform elastically, that is the

More information

A NEW REFINED THEORY OF PLATES WITH TRANSVERSE SHEAR DEFORMATION FOR MODERATELY THICK AND THICK PLATES

A NEW REFINED THEORY OF PLATES WITH TRANSVERSE SHEAR DEFORMATION FOR MODERATELY THICK AND THICK PLATES A NEW REFINED THEORY OF PLATES WITH TRANSVERSE SHEAR DEFORMATION FOR MODERATELY THICK AND THICK PLATES J.M. MARTÍNEZ VALLE Mechanics Department, EPS; Leonardo da Vinci Building, Rabanales Campus, Cordoba

More information

Elements of Continuum Elasticity. David M. Parks Mechanics and Materials II February 25, 2004

Elements of Continuum Elasticity. David M. Parks Mechanics and Materials II February 25, 2004 Elements of Continuum Elasticity David M. Parks Mechanics and Materials II 2.002 February 25, 2004 Solid Mechanics in 3 Dimensions: stress/equilibrium, strain/displacement, and intro to linear elastic

More information

1. Basic Operations Consider two vectors a (1, 4, 6) and b (2, 0, 4), where the components have been expressed in a given orthonormal basis.

1. Basic Operations Consider two vectors a (1, 4, 6) and b (2, 0, 4), where the components have been expressed in a given orthonormal basis. Questions on Vectors and Tensors 1. Basic Operations Consider two vectors a (1, 4, 6) and b (2, 0, 4), where the components have been expressed in a given orthonormal basis. Compute 1. a. 2. The angle

More information

CONTINUUM MECHANICS. lecture notes 2003 jp dr.-ing. habil. ellen kuhl technical university of kaiserslautern

CONTINUUM MECHANICS. lecture notes 2003 jp dr.-ing. habil. ellen kuhl technical university of kaiserslautern CONTINUUM MECHANICS lecture notes 2003 jp dr.-ing. habil. ellen kuhl technical university of kaiserslautern Contents Tensor calculus. Tensor algebra.................................... Vector algebra.................................

More information

Couple stress based strain gradient theory for elasticity

Couple stress based strain gradient theory for elasticity International Journal of Solids and Structures 39 (00) 731 743 www.elsevier.com/locate/ijsolstr Couple stress based strain gradient theory for elasticity F. Yang a,b, A.C.M. Chong a, D.C.C. Lam a, *, P.

More information

Mathematical Background

Mathematical Background CHAPTER ONE Mathematical Background This book assumes a background in the fundamentals of solid mechanics and the mechanical behavior of materials, including elasticity, plasticity, and friction. A previous

More information

CH.4. STRESS. Continuum Mechanics Course (MMC)

CH.4. STRESS. Continuum Mechanics Course (MMC) CH.4. STRESS Continuum Mechanics Course (MMC) Overview Forces Acting on a Continuum Body Cauchy s Postulates Stress Tensor Stress Tensor Components Scientific Notation Engineering Notation Sign Criterion

More information

Physics of Continuous media

Physics of Continuous media Physics of Continuous media Sourendu Gupta TIFR, Mumbai, India Classical Mechanics 2012 October 26, 2012 Deformations of continuous media If a body is deformed, we say that the point which originally had

More information

Mathematical model of static deformation of micropolar elastic circular thin bar

Mathematical model of static deformation of micropolar elastic circular thin bar Mathematical model of static deformation of micropolar elastic circular thin bar Mathematical model of static deformation of micropolar elastic circular thin bar Samvel H. Sargsyan, Meline V. Khachatryan

More information

Draft:S Ghorai. 1 Body and surface forces. Stress Principle

Draft:S Ghorai. 1 Body and surface forces. Stress Principle Stress Principle Body and surface forces Stress is a measure of force intensity, either within or on the bounding surface of a body subjected to loads. It should be noted that in continuum mechanics a

More information

Fundamentals of Fluid Dynamics: Elementary Viscous Flow

Fundamentals of Fluid Dynamics: Elementary Viscous Flow Fundamentals of Fluid Dynamics: Elementary Viscous Flow Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research

More information

FINAL EXAMINATION. (CE130-2 Mechanics of Materials)

FINAL EXAMINATION. (CE130-2 Mechanics of Materials) UNIVERSITY OF CLIFORNI, ERKELEY FLL SEMESTER 001 FINL EXMINTION (CE130- Mechanics of Materials) Problem 1: (15 points) pinned -bar structure is shown in Figure 1. There is an external force, W = 5000N,

More information

CRACK-TIP DRIVING FORCE The model evaluates the eect of inhomogeneities by nding the dierence between the J-integral on two contours - one close to th

CRACK-TIP DRIVING FORCE The model evaluates the eect of inhomogeneities by nding the dierence between the J-integral on two contours - one close to th ICF 100244OR Inhomogeneity eects on crack growth N. K. Simha 1,F.D.Fischer 2 &O.Kolednik 3 1 Department ofmechanical Engineering, University of Miami, P.O. Box 248294, Coral Gables, FL 33124-0624, USA

More information

Cellular solid structures with unbounded thermal expansion. Roderic Lakes. Journal of Materials Science Letters, 15, (1996).

Cellular solid structures with unbounded thermal expansion. Roderic Lakes. Journal of Materials Science Letters, 15, (1996). 1 Cellular solid structures with unbounded thermal expansion Roderic Lakes Journal of Materials Science Letters, 15, 475-477 (1996). Abstract Material microstructures are presented which can exhibit coefficients

More information

DAMPING OF GENERALIZED THERMO ELASTIC WAVES IN A HOMOGENEOUS ISOTROPIC PLATE

DAMPING OF GENERALIZED THERMO ELASTIC WAVES IN A HOMOGENEOUS ISOTROPIC PLATE Materials Physics and Mechanics 4 () 64-73 Received: April 9 DAMPING OF GENERALIZED THERMO ELASTIC WAVES IN A HOMOGENEOUS ISOTROPIC PLATE R. Selvamani * P. Ponnusamy Department of Mathematics Karunya University

More information

σ = F/A. (1.2) σ xy σ yy σ zy , (1.3) σ xz σ yz σ zz The use of the opposite convention should cause no problem because σ ij = σ ji.

σ = F/A. (1.2) σ xy σ yy σ zy , (1.3) σ xz σ yz σ zz The use of the opposite convention should cause no problem because σ ij = σ ji. Cambridge Universit Press 978-0-521-88121-0 - Metal Forming: Mechanics Metallurg, Third Edition Ecerpt 1 Stress Strain An understing of stress strain is essential for analzing metal forming operations.

More information

Iranian Journal of Mathematical Sciences and Informatics Vol.2, No.2 (2007), pp 1-16

Iranian Journal of Mathematical Sciences and Informatics Vol.2, No.2 (2007), pp 1-16 Iranian Journal of Mathematical Sciences and Informatics Vol.2, No.2 (2007), pp 1-16 THE EFFECT OF PURE SHEAR ON THE REFLECTION OF PLANE WAVES AT THE BOUNDARY OF AN ELASTIC HALF-SPACE W. HUSSAIN DEPARTMENT

More information

σ = F/A. (1.2) σ xy σ yy σ zx σ xz σ yz σ, (1.3) The use of the opposite convention should cause no problem because σ ij = σ ji.

σ = F/A. (1.2) σ xy σ yy σ zx σ xz σ yz σ, (1.3) The use of the opposite convention should cause no problem because σ ij = σ ji. Cambridge Universit Press 978-1-107-00452-8 - Metal Forming: Mechanics Metallurg, Fourth Edition Ecerpt 1 Stress Strain An understing of stress strain is essential for the analsis of metal forming operations.

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING CAMBRIDGE, MASSACHUSETTS 02139

MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING CAMBRIDGE, MASSACHUSETTS 02139 MASSACHUSETTS NSTTUTE OF TECHNOLOGY DEPARTMENT OF MATERALS SCENCE AND ENGNEERNG CAMBRDGE, MASSACHUSETTS 39 3. MECHANCAL PROPERTES OF MATERALS PROBLEM SET SOLUTONS Reading Ashby, M.F., 98, Tensors: Notes

More information

EFFECT OF COUPLE-STRESS ON THE REFLECTION AND TRANSMISSION OF PLANE WAVES AT AN INTERFACE

EFFECT OF COUPLE-STRESS ON THE REFLECTION AND TRANSMISSION OF PLANE WAVES AT AN INTERFACE International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol., Issue, pp-5-8 ISSN: 9-665 EFFET OF OUPLE-STRESS ON THE REFLETION AND TRANSMISSION OF PLANE WAVES AT AN INTERFAE Mahabir

More information

Effect of Couple Stresses on the MHD of a Non-Newtonian Unsteady Flow between Two Parallel Porous Plates

Effect of Couple Stresses on the MHD of a Non-Newtonian Unsteady Flow between Two Parallel Porous Plates Effect of Couple Stresses on the MHD of a Non-Newtonian Unsteady Flow between Two Parallel Porous Plates N. T. M. Eldabe A. A. Hassan and Mona A. A. Mohamed Department of Mathematics Faculty of Education

More information

Solutions for Fundamentals of Continuum Mechanics. John W. Rudnicki

Solutions for Fundamentals of Continuum Mechanics. John W. Rudnicki Solutions for Fundamentals of Continuum Mechanics John W. Rudnicki December, 015 ii Contents I Mathematical Preliminaries 1 1 Vectors 3 Tensors 7 3 Cartesian Coordinates 9 4 Vector (Cross) Product 13 5

More information

Mathematical Tripos Part IA Lent Term Example Sheet 1. Calculate its tangent vector dr/du at each point and hence find its total length.

Mathematical Tripos Part IA Lent Term Example Sheet 1. Calculate its tangent vector dr/du at each point and hence find its total length. Mathematical Tripos Part IA Lent Term 205 ector Calculus Prof B C Allanach Example Sheet Sketch the curve in the plane given parametrically by r(u) = ( x(u), y(u) ) = ( a cos 3 u, a sin 3 u ) with 0 u

More information

Non-Classical Continuum Theories for Solid and Fluent Continua. Aaron Joy

Non-Classical Continuum Theories for Solid and Fluent Continua. Aaron Joy Non-Classical Continuum Theories for Solid and Fluent Continua By Aaron Joy Submitted to the graduate degree program in Mechanical Engineering and the Graduate Faculty of the University of Kansas in partial

More information

Exercise: concepts from chapter 5

Exercise: concepts from chapter 5 Reading: Fundamentals of Structural Geology, Ch 5 1) Study the oöids depicted in Figure 1a and 1b. Figure 1a Figure 1b Figure 1. Nearly undeformed (1a) and significantly deformed (1b) oöids with spherulitic

More information

Chapter 3: Stress and Equilibrium of Deformable Bodies

Chapter 3: Stress and Equilibrium of Deformable Bodies Ch3-Stress-Equilibrium Page 1 Chapter 3: Stress and Equilibrium of Deformable Bodies When structures / deformable bodies are acted upon by loads, they build up internal forces (stresses) within them to

More information

SOME PROBLEMS OF THERMOELASTIC EQUILIBRIUM OF A RECTANGULAR PARALLELEPIPED IN TERMS OF ASYMMETRIC ELASTICITY

SOME PROBLEMS OF THERMOELASTIC EQUILIBRIUM OF A RECTANGULAR PARALLELEPIPED IN TERMS OF ASYMMETRIC ELASTICITY Georgian Mathematical Journal Volume 8 00) Number 4 767 784 SOME PROBLEMS OF THERMOELASTIC EQUILIBRIUM OF A RECTANGULAR PARALLELEPIPED IN TERMS OF ASYMMETRIC ELASTICITY N. KHOMASURIDZE Abstract. An effective

More information

Bending of Simply Supported Isotropic and Composite Laminate Plates

Bending of Simply Supported Isotropic and Composite Laminate Plates Bending of Simply Supported Isotropic and Composite Laminate Plates Ernesto Gutierrez-Miravete 1 Isotropic Plates Consider simply a supported rectangular plate of isotropic material (length a, width b,

More information

16.21 Techniques of Structural Analysis and Design Spring 2003 Unit #5 - Constitutive Equations

16.21 Techniques of Structural Analysis and Design Spring 2003 Unit #5 - Constitutive Equations 6.2 Techniques of Structural Analysis and Design Spring 2003 Unit #5 - Constitutive quations Constitutive quations For elastic materials: If the relation is linear: Û σ ij = σ ij (ɛ) = ρ () ɛ ij σ ij =

More information

Theory of Plasticity. Lecture Notes

Theory of Plasticity. Lecture Notes Theory of Plasticity Lecture Notes Spring 2012 Contents I Theory of Plasticity 1 1 Mechanical Theory of Plasticity 2 1.1 Field Equations for A Mechanical Theory.................... 2 1.1.1 Strain-displacement

More information

Reflection of quasi-p and quasi-sv waves at the free and rigid boundaries of a fibre-reinforced medium

Reflection of quasi-p and quasi-sv waves at the free and rigid boundaries of a fibre-reinforced medium Sādhan ā Vol. 7 Part 6 December 00 pp. 63 630. Printed in India Reflection of quasi-p and quasi-sv waves at the free and rigid boundaries of a fibre-reinforced medium A CHATTOPADHYAYRLKVENKATESWARLU and

More information

Elasticity. Jim Emery 1/10/97, 14:20. 1 Stress 2. 2 Strain 2. 3 The General Infinitesimal Displacement 3. 4 Shear Strain Notation In Various Books 4

Elasticity. Jim Emery 1/10/97, 14:20. 1 Stress 2. 2 Strain 2. 3 The General Infinitesimal Displacement 3. 4 Shear Strain Notation In Various Books 4 Elasticity Jim Emery 1/1/97, 14:2 Contents 1 Stress 2 2 Strain 2 3 The General Infinitesimal Displacement 3 4 Shear Strain Notation In Various Books 4 5 Vector Notation For Stress and Strain 4 6 The Elastic

More information

THE REFLECTION PHENOMENA OF SV-WAVES IN A GENERALIZED THERMOELASTIC MEDIUM

THE REFLECTION PHENOMENA OF SV-WAVES IN A GENERALIZED THERMOELASTIC MEDIUM Internat. J. Math. & Math. Sci. Vol., No. 8 () 59 56 S67 Hindawi Publishing Corp. THE REFLECTION PHENOMENA OF SV-WAVES IN A GENERALIZED THERMOELASTIC MEDIUM ABO-EL-NOUR N. ABD-ALLA and AMIRA A. S. AL-DAWY

More information

Mohr's Circle for 2-D Stress Analysis

Mohr's Circle for 2-D Stress Analysis Mohr's Circle for 2-D Stress Analysis If you want to know the principal stresses and maximum shear stresses, you can simply make it through 2-D or 3-D Mohr's cirlcles! You can know about the theory of

More information

7.6 Stress in symmetrical elastic beam transmitting both shear force and bending moment

7.6 Stress in symmetrical elastic beam transmitting both shear force and bending moment 7.6 Stress in symmetrical elastic beam transmitting both shear force and bending moment à It is more difficult to obtain an exact solution to this problem since the presence of the shear force means that

More information

COMPUTATIONAL ELASTICITY

COMPUTATIONAL ELASTICITY COMPUTATIONAL ELASTICITY Theory of Elasticity and Finite and Boundary Element Methods Mohammed Ameen Alpha Science International Ltd. Harrow, U.K. Contents Preface Notation vii xi PART A: THEORETICAL ELASTICITY

More information

PART 2: INTRODUCTION TO CONTINUUM MECHANICS

PART 2: INTRODUCTION TO CONTINUUM MECHANICS 7 PART : INTRODUCTION TO CONTINUUM MECHANICS In the following sections we develop some applications of tensor calculus in the areas of dynamics, elasticity, fluids and electricity and magnetism. We begin

More information

A hierarchy of higher order and higher grade continua Application to the plasticity and fracture of metallic foams

A hierarchy of higher order and higher grade continua Application to the plasticity and fracture of metallic foams A hierarchy of higher order and higher grade continua Application to the plasticity and fracture of metallic foams Samuel Forest Centre des Matériaux/UMR 7633 Mines Paris ParisTech /CNRS BP 87, 91003 Evry,

More information

(MPa) compute (a) The traction vector acting on an internal material plane with normal n ( e1 e

(MPa) compute (a) The traction vector acting on an internal material plane with normal n ( e1 e EN10: Continuum Mechanics Homework : Kinetics Due 1:00 noon Friday February 4th School of Engineering Brown University 1. For the Cauchy stress tensor with components 100 5 50 0 00 (MPa) compute (a) The

More information

Part 5 ACOUSTIC WAVE PROPAGATION IN ANISOTROPIC MEDIA

Part 5 ACOUSTIC WAVE PROPAGATION IN ANISOTROPIC MEDIA Part 5 ACOUSTIC WAVE PROPAGATION IN ANISOTROPIC MEDIA Review of Fundamentals displacement-strain relation stress-strain relation balance of momentum (deformation) (constitutive equation) (Newton's Law)

More information

Computational models of diamond anvil cell compression

Computational models of diamond anvil cell compression UDC 519.6 Computational models of diamond anvil cell compression A. I. Kondrat yev Independent Researcher, 5944 St. Alban Road, Pensacola, Florida 32503, USA Abstract. Diamond anvil cells (DAC) are extensively

More information

VYSOKÁ ŠKOLA BÁŇSKÁ TECHNICKÁ UNIVERZITA OSTRAVA

VYSOKÁ ŠKOLA BÁŇSKÁ TECHNICKÁ UNIVERZITA OSTRAVA VYSOKÁ ŠKOLA BÁŇSKÁ TECHNICKÁ UNIVERZITA OSTRAVA FAKULTA METALURGIE A MATERIÁLOVÉHO INŽENÝRSTVÍ APPLIED MECHANICS Study Support Leo Václavek Ostrava 2015 Title:Applied Mechanics Code: Author: doc. Ing.

More information

1. To analyze the deformation of a conical membrane, it is proposed to use a two-dimensional conical-polar coordinate system ( s,

1. To analyze the deformation of a conical membrane, it is proposed to use a two-dimensional conical-polar coordinate system ( s, EN2210: Continuum Mechanics Homework 2: Kinematics Due Wed September 26, 2012 School of Engineering Brown University 1. To analyze the deformation of a conical membrane, it is proposed to use a two-dimensional

More information

Chapter 2: Fluid Dynamics Review

Chapter 2: Fluid Dynamics Review 7 Chapter 2: Fluid Dynamics Review This chapter serves as a short review of basic fluid mechanics. We derive the relevant transport equations (or conservation equations), state Newton s viscosity law leading

More information

Research Article Dispersion of Love Waves in a Composite Layer Resting on Monoclinic Half-Space

Research Article Dispersion of Love Waves in a Composite Layer Resting on Monoclinic Half-Space Applied Mathematics Volume 011, Article ID 71349, 9 pages doi:10.1155/011/71349 Research Article Dispersion of Love Waves in a Composite Layer Resting on Monoclinic Half-Space Sukumar Saha BAS Division,

More information

International Journal of Pure and Applied Mathematics Volume 58 No ,

International Journal of Pure and Applied Mathematics Volume 58 No , International Journal of Pure and Applied Mathematics Volume 58 No. 2 2010, 195-208 A NOTE ON THE LINEARIZED FINITE THEORY OF ELASTICITY Maria Luisa Tonon Department of Mathematics University of Turin

More information

Earth Deformation Homework 1

Earth Deformation Homework 1 Earth Deformation Homework 1 Michal Dichter October 7, 14 Problem 1 (T+S Problem -5) We assume the setup of Figure -4 from Turcotte and Schubert: We are given the following values: hcc = 5 km hsb = 7 km

More information

Two problems in finite elasticity

Two problems in finite elasticity University of Wollongong Research Online University of Wollongong Thesis Collection 1954-2016 University of Wollongong Thesis Collections 2009 Two problems in finite elasticity Himanshuki Nilmini Padukka

More information

Mechanics of Earthquakes and Faulting

Mechanics of Earthquakes and Faulting Mechanics of Earthquakes and Faulting www.geosc.psu.edu/courses/geosc508 Overview Milestones in continuum mechanics Concepts of modulus and stiffness. Stress-strain relations Elasticity Surface and body

More information

Strain Transformation equations

Strain Transformation equations Strain Transformation equations R. Chandramouli Associate Dean-Research SASTRA University, Thanjavur-613 401 Joint Initiative of IITs and IISc Funded by MHRD Page 1 of 8 Table of Contents 1. Stress transformation

More information

Stability analysis of graphite crystal lattice with moment interactions

Stability analysis of graphite crystal lattice with moment interactions Proc. of XXXIV Summer School "Advanced Problems in Mechanics", St.-Petersburg, Russia. 2007. (Accepted) Stability analysis of graphite crystal lattice with moment interactions Igor E. Berinskiy berigor@mail.ru

More information

Keywords: Elastic waves, Finite rotations, Areolar strain, Complex shear.

Keywords: Elastic waves, Finite rotations, Areolar strain, Complex shear. Blucher Mechanical Engineering Proceedings May 14 vol. 1 num. 1 www.proceedings.blucher.com.br/evento/1wccm ELASTIC WAVES WITH FINITE ROTATIONS I. D. Kotchergenko Kotchergenko Engenharia Ltda. Belo Horizonte

More information

Mechanical Design in Optical Engineering

Mechanical Design in Optical Engineering Torsion Torsion: Torsion refers to the twisting of a structural member that is loaded by couples (torque) that produce rotation about the member s longitudinal axis. In other words, the member is loaded

More information

**********************************************************************

********************************************************************** Department of Civil and Environmental Engineering School of Mining and Petroleum Engineering 3-33 Markin/CNRL Natural Resources Engineering Facility www.engineering.ualberta.ca/civil Tel: 780.492.4235

More information

Dynamics of Glaciers

Dynamics of Glaciers Dynamics of Glaciers McCarthy Summer School 01 Andy Aschwanden Arctic Region Supercomputing Center University of Alaska Fairbanks, USA June 01 Note: This script is largely based on the Physics of Glaciers

More information

L8. Basic concepts of stress and equilibrium

L8. Basic concepts of stress and equilibrium L8. Basic concepts of stress and equilibrium Duggafrågor 1) Show that the stress (considered as a second order tensor) can be represented in terms of the eigenbases m i n i n i. Make the geometrical representation

More information

Concept Question Comment on the general features of the stress-strain response under this loading condition for both types of materials

Concept Question Comment on the general features of the stress-strain response under this loading condition for both types of materials Module 5 Material failure Learning Objectives review the basic characteristics of the uni-axial stress-strain curves of ductile and brittle materials understand the need to develop failure criteria for

More information

Elastic Fields of Dislocations in Anisotropic Media

Elastic Fields of Dislocations in Anisotropic Media Elastic Fields of Dislocations in Anisotropic Media a talk given at the group meeting Jie Yin, David M. Barnett and Wei Cai November 13, 2008 1 Why I want to give this talk Show interesting features on

More information

PROPAGATION OF A MODE-I CRACK UNDER THE IRWIN AND KHRISTIANOVICH BARENBLATT CRITERIA

PROPAGATION OF A MODE-I CRACK UNDER THE IRWIN AND KHRISTIANOVICH BARENBLATT CRITERIA Materials Science, Vol. 39, No. 3, 3 PROPAGATION OF A MODE-I CRACK UNDER THE IRWIN AND KHRISTIANOVICH BARENBLATT CRITERIA I. I. Argatov, M. Bach, and V. A. Kovtunenko UDC 539.3 We study the problem of

More information

. D CR Nomenclature D 1

. D CR Nomenclature D 1 . D CR Nomenclature D 1 Appendix D: CR NOMENCLATURE D 2 The notation used by different investigators working in CR formulations has not coalesced, since the topic is in flux. This Appendix identifies the

More information

ACCURATE FREE VIBRATION ANALYSIS OF POINT SUPPORTED MINDLIN PLATES BY THE SUPERPOSITION METHOD

ACCURATE FREE VIBRATION ANALYSIS OF POINT SUPPORTED MINDLIN PLATES BY THE SUPERPOSITION METHOD Journal of Sound and Vibration (1999) 219(2), 265 277 Article No. jsvi.1998.1874, available online at http://www.idealibrary.com.on ACCURATE FREE VIBRATION ANALYSIS OF POINT SUPPORTED MINDLIN PLATES BY

More information

Chapter 1. Continuum mechanics review. 1.1 Definitions and nomenclature

Chapter 1. Continuum mechanics review. 1.1 Definitions and nomenclature Chapter 1 Continuum mechanics review We will assume some familiarity with continuum mechanics as discussed in the context of an introductory geodynamics course; a good reference for such problems is Turcotte

More information

Linear Constitutive Relations in Isotropic Finite Viscoelasticity

Linear Constitutive Relations in Isotropic Finite Viscoelasticity Journal of Elasticity 55: 73 77, 1999. 1999 Kluwer Academic Publishers. Printed in the Netherlands. 73 Linear Constitutive Relations in Isotropic Finite Viscoelasticity R.C. BATRA and JANG-HORNG YU Department

More information

Shear stresses around circular cylindrical openings

Shear stresses around circular cylindrical openings Shear stresses around circular cylindrical openings P.C.J. Hoogenboom 1, C. van Weelden 1, C.B.M. Blom 1, 1 Delft University of Technology, the Netherlands Gemeentewerken Rotterdam, the Netherlands In

More information

You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.

You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator. MATHEMATICAL TRIPOS Part III Thursday 1 June 2006 1.30 to 4.30 PAPER 76 NONLINEAR CONTINUUM MECHANICS Attempt FOUR questions. There are SIX questions in total. The questions carry equal weight. STATIONERY

More information

Part 1 ELEMENTS OF CONTINUUM MECHANICS

Part 1 ELEMENTS OF CONTINUUM MECHANICS Part 1 ELEMENTS OF CONTINUUM MECHANICS CHAPTER 1 TENSOR PRELIMINARIES 1.1. Vectors An orthonormal basis for the three-dimensional Euclidean vector space is a set of three orthogonal unit vectors. The scalar

More information

On some exponential decay estimates for porous elastic cylinders

On some exponential decay estimates for porous elastic cylinders Arch. Mech., 56, 3, pp. 33 46, Warszawa 004 On some exponential decay estimates for porous elastic cylinders S. CHIRIȚĂ Faculty of Mathematics, University of Iaşi, 6600 Iaşi, Romania. In this paper we

More information

! EN! EU! NE! EE.! ij! NN! NU! UE! UN! UU

! EN! EU! NE! EE.! ij! NN! NU! UE! UN! UU A-1 Appendix A. Equations for Translating Between Stress Matrices, Fault Parameters, and P-T Axes Coordinate Systems and Rotations We use the same right-handed coordinate system as Andy Michael s program,

More information