# 3D and Planar Constitutive Relations

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 3D and Planar Constitutive Relations A School on Mechanics of Fibre Reinforced Polymer Composites Knowledge Incubation for TEQIP Indian Institute of Technology Kanpur PM Mohite Department of Aerospace Engineering Indian Institute of Technology Kanpur -5 January 017

2 3D Constitutive Relations

3 Generalized Hooke s Law: Generalized Hooke s law of the proportionality of stress and strain: Each of the six component of the stress at any point is a linear function of the six components of strain at that point. Concept of initial state Loading under two situations: - Isothermal and Reversible; - Adiabatic and Reversible. Stress components are the partial differential coefficients of a function (W) of the strain-components. 1 W W σ ij = + ε ij ε ji

4 Generalized Hooke s Law: Form of the Strain Energy Density Function (W): Homogeneous quadratic function of the strain components. 1 W = constant + C ij ε ij + C ijkl ε ij ε kl W is invariant Cij and Cijkl are tensors. W is taken to be zero when body is in the initial state in which Then, constant is zero. ε ij are zero.

5 Generalized Hooke s Law: For unstrained and unstressed body, Cij are zero. This leads to 1 W = Cijkl ε ij ε kl and σ ij = C ijkl ε kl i, j, k, l = 1,, 3 Cijkl is a fourth order (stiffness) tensor/matrix. (3 ) 4 = 81 independent constants!

6 Generalized Hooke s Law: Simply, you can view this as if you have a vector of 9 stress components which is related to a vector of 9 strain components through a matrix of 9x9! {σ }9 1 = [C ]9 9 {ε }9 1 C1111 C111 C1311 C111 [C] = C11 C311 C 3111 C311 C3311 C111 C1113 C111 C11 C113 C1131 C113 C1133 C11 C113 C11 C1 C13 C131 C13 C133 C131 C1313 C131 C13 C133 C1331 C133 C1333 C11 C113 C11 C1 C13 C131 C13 C133 C1 C13 C1 C C3 C31 C3 C33 C31 C313 C31 C3 C33 C331 C33 C333 C311 C3113 C311 C31 C313 C3131 C313 C3133 C31 C313 C31 C3 C33 C331 C33 C333 C331 C3313 C331 C33 C333 C3331 C333 C3333

7 Stress Tensor Symmetry: Stress symmetry: σ ij = σ ji σ ij = C ijkl ε kl σ ji = C jikl ε kl σ ij σ ji = 0 ( Cijkl C jikl ) ε kl = 0 C ijkl = C jikl six independent ways to express when i and j are taken together and still 9 ways to express k and l taken together. 6 9 = 54 independent constants!

8 Stress Tensor Symmetry: Simply, you can view this as if you have a vector of 6 stress components which is related to a vector of 9 strain components through a matrix of 6x9! {σ }6 1 = [C ]6 9 {ε }9 1 C1111 C 11 C3311 [C] = C 311 C 1311 C111 C111 C1 C331 C31 C1113 C13 C3313 C313 C111 C1 C331 C31 C11 C C33 C3 C113 C3 C333 C33 C1131 C31 C3331 C331 C113 C3 C333 C33 C131 C1313 C131 C13 C133 C1331 C133 C11 C113 C11 C1 C13 C131 C13 C1133 C33 C3333 C333 C1333 C133

9 Stress and Strain Tensor Symmetry: Strain symmetry: ε ij = ε ji σ ij = C ijkl ε kl σ ij = C ijlk ε lk σ ij σ ij = 0 ( Cijkl Cijlk ) ε kl = 0 C ijkl = C ijlk six independent ways to express when i and j are taken together and 6 ways to express k and l taken together. 6 6 = 36 independent constants!

10 Stress and Strain Tensor Symmetry: Or simply, you can view this as if you have a vector of 6 stress components which is related to a vector of 6 strain components through a matrix of 6x6! {σ }6 1 = [C ]6 6 {ε }6 1 C1111 C 11 C3311 [C] = C 311 C 1311 C111 C11 C C33 C3 C1133 C33 C3333 C333 C113 C3 C333 C33 C1113 C13 C3313 C31 C13 C1333 C133 C1313 C1 C133 C13 C11 C111 C1 C331 C31 C131 C11

11 Stress and Strain Tensor Symmetry: In other words, σ11 C1111 σ C 11 σ33 C3311 = σ3 C311 σ13 C1311 σ1 C111 C11 C1133 C113 C1113 C111 ε11 C C33 C3 C13 C1 ε C33 C3333 C333 C3313 C331 ε33 C3 C333 C33 C31 C31 ε3 C13 C1333 C133 C1313 C131 ε13 C1 C133 C13 C11 C11 ε1

12 Voigt Notation: Using Voigt notation - a way to represent a symmetric tensor by reducing its order For stress components: {σ σ11 σ1 σ13 1 σ σ 3 4 σ33 3 σ σ33 σ3 σ13 σ1} ={σ1 σ σ3 σ4 σ5 σ6} Strain Components: {ε11 ε ε33 ε3 ε13 ε1} ={ε1 ε ε3 ε4 ε5 ε6}

13 Stress and Strain Tensor Symmetries: Instead of writing C as a fourth order tensor, written as a second order tensor and stress and strains tensors are written as vectors! σ1 C11 σ C 1 σ3 C31 = σ4 C41 σ5 C51 σ6 C61 C1 C13 C14 C15 C16 ε1 C C3 C4 C5 C6 ε C3 C33 C34 C35 C36 ε3 C4 C43 C44 C45 C46 ε4 C51 C53 C54 C55 C56 ε5 C6 C63 C64 C65 C66 ε6

14 Existence of W: Existence of W : Hyperelastic materials Invariant Positive Definite 1 W = C ij ε j ε i 1 W = C ij ε i ε j ( Cij C ji ) ε iε j = 0 C ij = C ji

15 Existence of W: σ1 C11 C1 C13 σ C C 3 σ3 C33 = σ4 σ5 σ6 C14 C15 C16 ε1 C4 C5 C6 ε C34 C35 C36 ε3 C44 C45 C46 ε4 C55 C56 ε5 C66 ε6 1 independent constants!

16 Symmetries: Stress symmetry C ijkl = C jikl Minor Symmetries Strain symmetry C ijkl = C jikl Existence of W: C ijkl = C klij Major Symmetry

17 Transformations: Transformations Prime denotes the transformed coordinates. aij denotes the components of a transformation matrix

18 Material Symmetry: Further reduction in constants obtained by material symmetry Symmetry Definition: Any geometrical figure which can be brought to coincidence with itself, by an operation which changes the position of any of its points, is said to possess symmetry. Rotation and Reflection

19 Form of W: Quadratic in strain components: Note that the strains used are engineering strains.

20 Material Symmetry: One Plane of Material Symmetry: Monoclinic Materials

21 Material Symmetry: One Plane of Material Symmetry Transformation of axes: Transformation matrix: Transformation of strains:

22 Material Symmetry: One Plane of Material Symmetry Transformation of stresses:

23 Material Symmetry: One Plane of Material Symmetry Transformation of stiffness: C ij' = C ij Comparison of stress components: C14 = C15 = 0

24 Material Symmetry: One Plane of Material Symmetry Similarly, 13 independent constants

25 Material Symmetry: One Plane of Material Symmetry Second Approach: Invariance of W W for Hyperelastic material

26 Material Symmetry: One Plane of Material Symmetry Second Approach: Invariance of W For W to be invariant the product terms ε 1ε 4, ε 1ε 5, ε ε 4, ε ε 5, ε 3ε 4, ε 3ε 5, ε 4 ε 6, ε 5ε 6 must vanish, that is, C14, C15, C 4, C 5, C 34, C 35, C 46, C 56 are zero

27 Material Symmetry: Two Orthogonal Planes of Material Symmetry Two Orthogonal Planes of Material Symmetry: Orthotropic Materials Transformation of axes: Transformation matrix: Transformation of strains:

28 Material Symmetry: Two Orthogonal Planes of Material Symmetry Transformation of stresses: Transformation of stiffness: C ij' = C ij

29 Material Symmetry: Two Orthogonal Planes of Material Symmetry Comparison of stresses: C16 = 0 Similarly,

30 Material Symmetry: Two Orthogonal Planes of Material Symmetry Stiffness Tensor: 9 independent constants

31 Material Symmetry: Two Orthogonal Planes of Material Symmetry Second approach: Invariance of w W for monoclinic material: C11ε 1 + C ε + C33ε 3 + C 44 ε C55ε 5 + C 66ε 6 + C1ε 1ε + C13ε 1ε 3 + W = C16ε 1ε 6 + C 3ε ε 3 + C 6 ε ε 6 + C36ε 3ε 6 C 45ε 4 ε 5 or form of W: ε 1, ε, ε 3, ε 4, ε 5, ε 6, W = W ε 1ε, ε 1ε 3, ε 1ε 6, ε ε, ε ε, ε ε, ε ε

32 Material Symmetry: Two Orthogonal Planes of Material Symmetry For W to be invariant under the strain transformations the product terms ε 1ε 6, ε ε 6, ε 3ε 6, ε 4 ε 5 must vanish, which is possible when C16, C 6, C36, C 45 are zero W for orthotropic material: 1 C11ε 1 + C ε + C33ε 3 + C 44 ε 4 + C55ε 5 + C 66ε 6 W = C1 ε 1ε + C13ε 1ε 3 + C 3ε ε 3 +

33 Material Symmetry: Two Orthogonal Planes of Material Symmetry When material has two orthogonal planes of symmetry then it also symmetric about a plane which is mutually orthogonal earlier two planes! Such materials are called Orthotropic Materials. Select now the remaining plane x1-x3 as the third orthogonal plane of material symmetry in addition to earlier two planes Follow the same procedure, either comparing the stresses or invariance of W There will be no change in the final stiffness tensor. Number of independent constants will be still 9!

34 Material Symmetry: Two Orthogonal Planes of Material Symmetry When material has two orthogonal planes of symmetry then it also symmetric about a plane which is mutually orthogonal earlier two planes! Alternately: Now select the plane x1-x3 as the second orthogonal plane of material symmetry in addition to x1-x plane

35 Isotropy in a Plane: Isotropic behaviour of UD lamina in the cross-sectional plane (perpendicular to fibre s length) Transformation matrix:

36 Isotropy in a Plane: Transformation of strains:

37 Material Symmetry: Two Orthogonal Planes of Material Symmetry W for monoclinic material: ε 1, ε, ε 3, ε 4, ε 5, ε 6, W = W ε 1ε, ε 1ε 3, ε 1ε 6, ε ε, ε ε, ε ε, ε ε For invariance ε 1ε 6, ε ε 6, ε 3ε 6, ε 4 ε 5 = 0 C16, C 6, C 36, C 45 are zero

38 Isotropy in a Plane: Trigonometric identities for the strains: ' ' ε + ε 33 = ε + ε 33, ( ) = (ε ) + (ε ) ε ε 33 ( ε 3 ) = (ε1 ) + ( ε 13 ) ' ε ' 1 ' + ε 33 ' ε 3 ' 13, Form of W: W = W ε11, ε + ε 33, ε ε 33 ( ε 3 ), ( ε 1 ) + ( ε 13 ) ) ' ' ' ' ' ' ε 33 W = W ε 11, ε + ε 33, ε ε 3 ( ( ),( ) + ( ) ' ε1 ' ε13

39 Isotropy in a Plane: Strain energy density function for orthotropic material: W = C11ε 11 + C1 ε 11ε + C13ε 11ε 33 + C ε + C 33ε 33 + C 3ε ε C 44 ε 3 + 4C 55ε C 66 ε 1 rearranging W = C11ε 11 + ε 11 ( C1 ε + C13ε 33 ) + 4C55ε C 66 ε C 33ε 33 + C 3ε ε C 44 ε 3 + C ε

40 Isotropy in a Plane: W = C11ε 11 + ε 11 ( C1 ε + C13ε 33 ) + 4C55ε C 66 ε C ε + C 33ε 33 + C 3ε ε C 44 ε 3 In the second bracket, we take C1 = C13 In the third bracket, we take C55 = C 66 C = C33 and C 3 + C 33ε 33 + C 3ε ε C 44ε 3 Rearrange the last bracket with C ε unchanged = C ( ε + ε 33 ) C ε ε 33 + C 3ε ε C 44 ε 3

41 Isotropy in a Plane: Rearrange the last bracket further as C ε + C 33ε 33 + C 3ε ε C 44 ε 3 = C ( ε + ε 33 ) we need C 44 ( C C 3 ) ε ε 33 C 44 ε 3 C C 3 =

42 Isotropy in a Plane: Stiffness tensor C11 C ij = C1 C1 0 0 C C C 0 0 C C 3 0 Sym C 66 5 independent constants Such materials are called Transversely isotropic materials. Define: where C 66

43 Transverse Isotropy with an Additional Orthogonal Plane: Consider isotropy in x1-x plane as well strain transformation

44 Transverse Isotropy with an Additional Orthogonal Plane: Trigonometric identities: ε 11 + ε = ' ε 11 ' + ε, ( ) = (ε ) + (ε ) ε 11ε ( ε 1 ) = (ε13 ) + ( ε 3 ) ' ε 11 ' + ε ' 13 ' ε 1 ' 3, Form of W: W = W ε11 + ε, ε 33, ε 11ε ( ε 1 ), ( ε13 ) + ( ε 3 ) ) ' ' ' ' ' ' ε ε1 W = W ε 11 + ε, ε 33, ε 11 ( ( ),( ) + ( ) ' ε13 ' ε 3

45 Transverse Isotropy with an Additional Orthogonal Plane: In the second bracket, we take C1 = C 3 In the third bracket, we take Rearrange the last bracket with C11 = C and C1 unchanged

46 Isotropy: Two independent constants! C11 Cij = C1 C1 0 0 C11 C1 0 0 C C11 C1 0 Define: where C11 C1 Sym C11 C1 0

47 3D Constitutive Relations: Quick Review Generalized Hooke s Law: 81 independent constants Stress tensor symmetry: 54 independent constants Strain tensor symmetry: 36 independent constants Existence of W (Hyperelastic/Aelotropic): 1 independent constants Existence of one plane of material symmetry: 13 independent constants Existence of two/three mutually perpendicular planes of symmetry: (Orthotropic Material) 9 independent constants One plane of isotropy: 5 independent constants Two/three/infinite planes of isotropy: independent constants

48 3D Constitutive Relations for Orthotropic Materials

49 Constitutive Relations for Orthotropic Materials: Strain-stress Relations Normal stresses and strains

50 Constitutive Relations for Orthotropic Materials: Shear stresses and strains Poisson s ratio: (no sum over i, j) In general, ν ij ν ji

51 Constitutive Relations for Orthotropic Materials: Determination of Engineering Constants:

52 Constitutive Relations for Orthotropic Materials: Matrix-vector form: where, Always work with compliance tensor. It is easy to remember.

53 Constitutive Relations for Orthotropic Materials: Important Relations: Reciprocal relation where,

54 Constitutive Relations for Orthotropic Materials: Stiffness Relations: where,

55 Constraints on Engineering Constants: For strain energy to be positive definite both Compliance and Stiffness tensors must be positive definite. Strain energy to be positive definite the diagonal entries of the Compliance tensor must be positive. Similarly, the diagonal entries of the Stiffness tensor must be positive and the determinant must also be positive

56 Constraints on Engineering Constants: Constraint on Poisson s ratio: From constraint on determinant:

57 Constraints on Engineering Constants: For transverse isotropic material: with we get, Finally, leads to the condition For isotropic materials: and

58 Constitutive Relations: Transformations 13 Principal material directions xyz global reference directions Transformation matrix for rotation about z-axis:

59 Constitutive Relations: Stress Transformations Stress transformation: ' σ ij For example, that is, Transformation matrix: = a ki aljσ kl

60 Constitutive Relations: Stress and Strain Transformations Transformation matrix: where, m=cosθ and n=sinθ

61 Constitutive Relations: Stress and Strain Transformations Stress transformation: ' ε ij For example, that is, Now using = a ki alj ε kl

62 Constitutive Relations: Stress and Strain Transformations Strain transformation: Transformation matrix:

63 Constitutive Relations: Stress and Strain Transformations Stress Transformation: Strain Transformation:

64 Constitutive Relations: Stiffness Transformations From the first principles: Writing in global coordinates leads to and

65 Constitutive Relations: Stiffness Transformations Results in monoclinic behaviour! Constitutive relations:

66 Constitutive Relations: Stiffness Transformations

67 Constitutive Relations: Compliance Transformations From the first principles: Writing in global coordinates Or and

68 Constitutive Relations: Compliance Transformations The transformed Stiffness and Compliance tensors are symmetric! From invariance of W one can show 1 [T1 ] T = [T ] and 1 [T ] T = [T1 ]

69 Constitutive Relations: Compliance Transformations

70 Constitutive Relations: Thermal Effects Coefficient of thermal expansion is different in 3 directions!

71 Constitutive Relations: Thermal Effects Thermal strain in principal material directions where, These strains will not produce stresses unless restricted! Transforming strains into global coordinates We get,

72 Constitutive Relations: Thermal Effects That is, where, and

73 Constitutive Relations: Thermo-Elastic Equations Total strains: Mechanical strains: Thus, gives the thermo-elastic constitutive equations as

74 Constitutive Relations: Thermo-Elastic Equations Stresses in global direction: where,

75 Constitutive Relations: Hygral Effects Hygral strains: where, are coefficient s of hygral expansion

76 Constitutive Relations: Hygro-Thermal Effects Total strains: with, The hygro-thermo-elastic constitutive equation:

77 D Constitutive Relations for Orthotropic Materials

78 Constitutive Relations: Planar Equations Constitutive relation in 3D: Transverse stresses are zero: σ zz = τ xz = τ yz = 0 γ yz = S 44τ yz + S 45τ xz = 0 leads to γ xz = S 45τ xz + S 55τ yz = 0

79 Constitutive Relations: Transverse Strain Transverse normal strain: ε zz = S 13σ xx + S 3σ yy + S 36τ xy 0

80 Constitutive Relations: In-Plane Stresses

81 Constitutive Relations: In-Plane Stresses

82 Constitutive Relations: Planar Equations Constitutive relation in 3D (Principal Directions): Transverse stresses are zero: leads to

83 Planar Relations: Principal Directions Transverse normal strain: ε 33 = S13σ 11 + S 3σ + S 36 τ 1 0 that is, ε 33 = S13σ 11 + S 3σ 0 Therefore, for planar case

84 Planar Relations: Principal Directions Transverse normal strain from stiffness relations: Transverse normal stress: gives

85 Planar Relations: Principal Directions Stresses in principal directions: Putting ε 33 in terms of ε 11, ε

86 Planar Relations: Principal Directions Stresses in principal directions: And can be written in a form as Qij and Cij are not same Inverse f orm: Sij are same as in 3D relations

87 Planar Relations: Principal Directions Reduced Stiffness Matrix: Stiffness Compliance in terms of engineering constants

88 Planar Relations: Principal Directions Stresses in principal directions:

89 Planar Relations: Principal Directions Stresses in principal directions:

90 Planar Relations: Transformation of Stresses and Strains Stresses in principal directions: Strains in principal directions:

91 Planar Relations: Transformation of Stresses and Strains Stresses in principal directions:

92 Planar Relations: Transformation of Stresses and Strains

93 Planar Relations: Transformation of Stresses and Strains Stresses in principal directions:

94 Planar Relations: Hygro-thermo-elastic Relations Stresses in principal directions: Stresses in principal directions Stresses in global directions:

95 Thank You

### Understand basic stress-strain response of engineering materials.

Module 3 Constitutive quations Learning Objectives Understand basic stress-strain response of engineering materials. Quantify the linear elastic stress-strain response in terms of tensorial quantities

### ELASTICITY (MDM 10203)

LASTICITY (MDM 10203) Lecture Module 5: 3D Constitutive Relations Dr. Waluyo Adi Siswanto University Tun Hussein Onn Malaysia Generalised Hooke's Law In one dimensional system: = (basic Hooke's law) Considering

### Continuum mechanics V. Constitutive equations. 1. Constitutive equation: definition and basic axioms

Continuum mechanics office Math 0.107 ales.janka@unifr.ch http://perso.unifr.ch/ales.janka/mechanics Mars 16, 2011, Université de Fribourg 1. Constitutive equation: definition and basic axioms Constitutive

### Tensor Visualization. CSC 7443: Scientific Information Visualization

Tensor Visualization Tensor data A tensor is a multivariate quantity Scalar is a tensor of rank zero s = s(x,y,z) Vector is a tensor of rank one v = (v x,v y,v z ) For a symmetric tensor of rank 2, its

### 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

DESIGN OF LAMINATES FOR IN-PLANOADING G. VERCHERY ISMANS 44 avenue F.A. Bartholdi, 72000 Le Mans, France Georges.Verchery@m4x.org SUMMARY This work relates to the design of laminated structures primarily

### Continuum Mechanics. Continuum Mechanics and Constitutive Equations

Continuum Mechanics Continuum Mechanics and Constitutive Equations Continuum mechanics pertains to the description of mechanical behavior of materials under the assumption that the material is a uniform

### 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

### MATERIAL ELASTIC ANISOTROPIC command

MATERIAL ELASTIC ANISOTROPIC command.. Synopsis The MATERIAL ELASTIC ANISOTROPIC command is used to specify the parameters associated with an anisotropic linear elastic material idealization. Syntax The

### Basic Concepts of Strain and Tilt. Evelyn Roeloffs, USGS June 2008

Basic Concepts of Strain and Tilt Evelyn Roeloffs, USGS June 2008 1 Coordinates Right-handed coordinate system, with positions along the three axes specified by x,y,z. x,y will usually be horizontal, and

### MODELING OF CONCRETE MATERIALS AND STRUCTURES. Kaspar Willam. Isotropic Elastic Models: Invariant vs Principal Formulations

MODELING OF CONCRETE MATERIALS AND STRUCTURES Kaspar Willam University of Colorado at Boulder Class Meeting #2: Nonlinear Elastic Models Isotropic Elastic Models: Invariant vs Principal Formulations Elastic

### Constitutive models: Incremental plasticity Drücker s postulate

Constitutive models: Incremental plasticity Drücker s postulate if consistency condition associated plastic law, associated plasticity - plastic flow law associated with the limit (loading) surface Prager

### Chapter 2 - Macromechanical Analysis of a Lamina. Exercise Set. 2.1 The number of independent elastic constants in three dimensions are: 2.

Chapter - Macromechanical Analysis of a Lamina Exercise Set. The number of independent elastic constants in three dimensions are: Anisotropic Monoclinic 3 Orthotropic 9 Transversely Orthotropic 5 Isotropic.

### 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

### Problem Set 2 Due Tuesday, September 27, ; p : 0. (b) Construct a representation using five d orbitals that sit on the origin as a basis: 1

Problem Set 2 Due Tuesday, September 27, 211 Problems from Carter: Chapter 2: 2a-d,g,h,j 2.6, 2.9; Chapter 3: 1a-d,f,g 3.3, 3.6, 3.7 Additional problems: (1) Consider the D 4 point group and use a coordinate

### TRESS - STRAIN RELATIONS

TRESS - STRAIN RELATIONS Stress Strain Relations: Hook's law, states that within the elastic limits the stress is proportional to t is impossible to describe the entire stress strain curve with simple

### Solid State Theory Physics 545

olid tate Theory hysics 545 Mechanical properties of materials. Basics. tress and strain. Basic definitions. Normal and hear stresses. Elastic constants. tress tensor. Young modulus. rystal symmetry and

### 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

### ME 582 Advanced Materials Science. Chapter 2 Macromechanical Analysis of a Lamina (Part 2)

ME 582 Advanced Materials Science Chapter 2 Macromechanical Analysis of a Lamina (Part 2) Laboratory for Composite Materials Research Department of Mechanical Engineering University of South Alabama, Mobile,

### 1 Stress and Strain. Introduction

1 Stress and Strain Introduction This book is concerned with the mechanical behavior of materials. The term mechanical behavior refers to the response of materials to forces. Under load, a material may

### Fracture Mechanics of Composites with Residual Thermal Stresses

J. A. Nairn Material Science & Engineering, University of Utah, Salt Lake City, Utah 84 Fracture Mechanics of Composites with Residual Thermal Stresses The problem of calculating the energy release rate

### Constitutive models: Incremental (Hypoelastic) Stress- Strain relations. and

Constitutive models: Incremental (Hypoelastic) Stress- Strain relations Example 5: an incremental relation based on hyperelasticity strain energy density function and 14.11.2007 1 Constitutive models:

### CONSTITUTIVE RELATIONS FOR LINEAR ELASTIC SOLIDS

Chapter 9 CONSTITUTIV RLATIONS FOR LINAR LASTIC SOLIDS Figure 9.1: Hooke memorial window, St. Helen s, Bishopsgate, City of London 211 212 CHAPTR 9. CONSTITUTIV RLATIONS FOR LINAR LASTIC SOLIDS 9.1 Mechanical

### Classical Mechanics. Luis Anchordoqui

1 Rigid Body Motion Inertia Tensor Rotational Kinetic Energy Principal Axes of Rotation Steiner s Theorem Euler s Equations for a Rigid Body Eulerian Angles Review of Fundamental Equations 2 Rigid body

### 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

### 2. Mechanics of Materials: Strain. 3. Hookes's Law

Mechanics of Materials Course: WB3413, Dredging Processes 1 Fundamental Theory Required for Sand, Clay and Rock Cutting 1. Mechanics of Materials: Stress 1. Introduction 2. Plane Stress and Coordinate

### MATERIAL MECHANICS, SE2126 COMPUTER LAB 4 MICRO MECHANICS. E E v E E E E E v E E + + = m f f. f f

MATRIAL MCHANICS, S226 COMPUTR LAB 4 MICRO MCHANICS 2 2 2 f m f f m T m f m f f m v v + + = + PART A SPHRICAL PARTICL INCLUSION Consider a solid granular material, a so called particle composite, shown

### Chapter 16: Elastic Solids

Chapter 16: Elastic Solids Chapter 16: Elastic Solids... 366 16.1 Introduction... 367 16.2 The Elastic Strain... 368 16.2.1 The displacement vector... 368 16.2.2 The deformation gradient... 368 16.2.3

### Chapter 3. Load and Stress Analysis

Chapter 3 Load and Stress Analysis 2 Shear Force and Bending Moments in Beams Internal shear force V & bending moment M must ensure equilibrium Fig. 3 2 Sign Conventions for Bending and Shear Fig. 3 3

### Modeling of Fiber-Reinforced Membrane Materials Daniel Balzani. (Acknowledgement: Anna Zahn) Tasks Week 2 Winter term 2014

Institute of echanics and Shell Structures Faculty Civil Engineering Chair of echanics odeling of Fiber-Reinforced embrane aterials OOC@TU9 Daniel Balani (Acknowledgement: Anna Zahn Tasks Week 2 Winter

### Composites Design and Analysis. Stress Strain Relationship

Composites Design and Analysis Stress Strain Relationship Composite design and analysis Laminate Theory Manufacturing Methods Materials Composite Materials Design / Analysis Engineer Design Guidelines

### Enhancing Prediction Accuracy In Sift Theory

18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS Enhancing Prediction Accuracy In Sift Theory J. Wang 1 *, W. K. Chiu 1 Defence Science and Technology Organisation, Fishermans Bend, Australia, Department

### EFFECT OF LAMINATION ANGLE AND THICKNESS ON ANALYSIS OF COMPOSITE PLATE UNDER THERMO MECHANICAL LOADING

Journal of MECHANICAL ENGINEERING Strojnícky časopis, VOL 67 (217), NO 1, 5-22 EFFECT OF LAMINATION ANGLE AND THICKNESS ON ANALYSIS OF COMPOSITE PLATE UNDER THERMO MECHANICAL LOADING Arnab Choudhury 1,

### 9 Strength Theories of Lamina

9 trength Theories of Lamina 9- TRENGTH O ORTHOTROPIC LAMINA or isotropic materials the simplest method to predict failure is to compare the applied stresses to the strengths or some other allowable stresses.

### Lectures on. Constitutive Modelling of Arteries. Ray Ogden

Lectures on Constitutive Modelling of Arteries Ray Ogden University of Aberdeen Xi an Jiaotong University April 2011 Overview of the Ingredients of Continuum Mechanics needed in Soft Tissue Biomechanics

### Lecture 7. Properties of Materials

MIT 3.00 Fall 2002 c W.C Carter 55 Lecture 7 Properties of Materials Last Time Types of Systems and Types of Processes Division of Total Energy into Kinetic, Potential, and Internal Types of Work: Polarization

### DESIGN OF COMPOSITE LAMINATED STRUCTURES BY POLAR METHOD AND TOPOLOGY OPTIMISATION

DESIGN OF COMPOSIE LAMINAED SUCUES BY POLA MEOD AND OPOLOGY OPIMISAION A. Jibawy,,3, C. Julien,,3, B. Desmorat,,4, (*), A. Vincenti UPMC Univ Paris 6, UM 79, Institut Jean Le ond d Alembert B.P. 6 4, place

### Module 5: Laminate Theory Lecture 19: Hygro -thermal Laminate Theory. Introduction: The Lecture Contains: Laminate Theory with Thermal Effects

Module 5: Laminate Theory Lecture 19: Hygro -thermal Laminate Theory Introduction: In this lecture we are going to develop the laminate theory with thermal and hygral effects. Then we will develop the

### Two-dimensional flow in a porous medium with general anisotropy

Two-dimensional flow in a porous medium with general anisotropy P.A. Tyvand & A.R.F. Storhaug Norwegian University of Life Sciences 143 Ås Norway peder.tyvand@umb.no 1 Darcy s law for flow in an isotropic

### Module 7: Micromechanics Lecture 29: Background of Concentric Cylinder Assemblage Model. Introduction. The Lecture Contains

Introduction In this lecture we are going to introduce a new micromechanics model to determine the fibrous composite effective properties in terms of properties of its individual phases. In this model

### MECHANICS OF MATERIALS. EQUATIONS AND THEOREMS

1 MECHANICS OF MATERIALS. EQUATIONS AND THEOREMS Version 2011-01-14 Stress tensor Definition of traction vector (1) Cauchy theorem (2) Equilibrium (3) Invariants (4) (5) (6) or, written in terms of principal

### 2.1 Strain energy functions for incompressible materials

Chapter 2 Strain energy functions The aims of constitutive theories are to develop mathematical models for representing the real behavior of matter, to determine the material response and in general, to

### 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,

### A synergistic damage mechanics approach to mechanical response of composite laminates with ply cracks

Article A synergistic damage mechanics approach to mechanical response of composite laminates with ply cracks JOURNAL OF COMPOSITE MATERIALS Journal of Composite Materials 0(0) 7! The Author(s) 0 Reprints

### MAE 323: Chapter 4. Plane Stress and Plane Strain. The Stress Equilibrium Equation

The Stress Equilibrium Equation As we mentioned in Chapter 2, using the Galerkin formulation and a choice of shape functions, we can derive a discretized form of most differential equations. In Structural

### In this section, thermoelasticity is considered. By definition, the constitutive relations for Gradθ. This general case

Section.. Thermoelasticity In this section, thermoelasticity is considered. By definition, the constitutive relations for F, θ, Gradθ. This general case such a material depend only on the set of field

### 2 Introduction to mechanics

21 Motivation Thermodynamic bodies are being characterized by two competing opposite phenomena, energy and entropy which some researchers in thermodynamics would classify as cause and chance or determinancy

### Calculation of Energy Release Rate in Mode I Delamination of Angle Ply Laminated Composites

Copyright c 2007 ICCES ICCES, vol.1, no.2, pp.61-67, 2007 Calculation of Energy Release Rate in Mode I Delamination of Angle Ply Laminated Composites K. Gordnian 1, H. Hadavinia 1, G. Simpson 1 and A.

### CIVL4332 L1 Introduction to Finite Element Method

CIVL L Introduction to Finite Element Method CIVL L Introduction to Finite Element Method by Joe Gattas, Faris Albermani Introduction The FEM is a numerical technique for solving physical problems such

### 1.1 Stress, strain, and displacement! wave equation

32 geophysics 3: introduction to seismology. Stress, strain, and displacement wave equation From the relationship between stress, strain, and displacement, we can derive a 3D elastic wave equation. Figure.

### Beam Models. Wenbin Yu Utah State University, Logan, Utah April 13, 2012

Beam Models Wenbin Yu Utah State University, Logan, Utah 843-4130 April 13, 01 1 Introduction If a structure has one of its dimensions much larger than the other two, such as slender wings, rotor blades,

### FEM Modeling of a 3D Printed Carbon Fiber Pylon

FEM Modeling of a 3D Printed Carbon Fiber Pylon I. López G.*, B. Chiné, and J.L. León S. Costa Rica Institute of Technology, School of Materials Science and Engineering, Cartago, Costa Rica *Corresponding

### Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods of Structural Analysis

uke University epartment of Civil and Environmental Engineering CEE 42L. Matrix Structural Analysis Henri P. Gavin Fall, 22 Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods

### Chemistry 431. NC State University. Lecture 17. Vibrational Spectroscopy

Chemistry 43 Lecture 7 Vibrational Spectroscopy NC State University The Dipole Moment Expansion The permanent dipole moment of a molecule oscillates about an equilibrium value as the molecule vibrates.

### MECHANICS OF MATERIALS

Third E CHAPTER 2 Stress MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University and Strain Axial Loading Contents Stress & Strain:

### Computational Biomechanics Lecture 2: Basic Mechanics 2. Ulli Simon, Frank Niemeyer, Martin Pietsch

Computational Biomechanics 016 Lecture : Basic Mechanics Ulli Simon, Frank Niemeyer, Martin Pietsch Scientific Computing Centre Ulm, UZWR Ulm University Contents .7 Static Equilibrium Important: Free-body

### THE MECHANICAL BEHAVIOR OF ORIENTED 3D FIBER STRUCTURES

Lappeenranta University of Technology School of Engineering Science Degree Program in Computational Engineering and Technical Physics Master s Thesis Alla Kliuzheva THE MECHANICAL BEHAVIOR OF ORIENTED

### EFFECT OF ELLIPTIC OR CIRCULAR HOLES ON THE STRESS DISTRIBUTION IN PLATES

EFFECT OF ELLIPTIC OR CIRCULAR HOLES ON THE STRESS DISTRIBUTION IN PLATES OF WOOD OR PLYWOOD CONSIDERED AS ORTHOTROPIC MATERIALS Information Revied and Reaffirmed March 1956 No. 1510 EFFECT OF ELLIPTIC

Chapter 5 CENTRIC TENSION OR COMPRESSION ( AXIAL LOADING ) 5.1 DEFINITION A construction member is subjected to centric (axial) tension or compression if in any cross section the single distinct stress

### Stresses and Strains in flexible Pavements

Stresses and Strains in flexible Pavements Multi Layered Elastic System Assumptions in Multi Layered Elastic Systems The material properties of each layer are homogeneous property at point A i is the same

### Calculating anisotropic physical properties from texture data using the MTEX open source package

Calculating anisotropic physical properties from texture data using the MTEX open source package David Mainprice 1, Ralf Hielscher 2, Helmut Schaeben 3 February 10, 2011 1 Geosciences Montpellier UMR CNRS

### SEISMOLOGY I. Laurea Magistralis in Physics of the Earth and of the Environment. Elasticity. Fabio ROMANELLI

SEISMOLOGY I Laurea Magistralis in Physics of the Earth and of the Environment Elasticity Fabio ROMANELLI Dept. Earth Sciences Università degli studi di Trieste romanel@dst.units.it 1 Elasticity and Seismic

### Plane and axisymmetric models in Mentat & MARC. Tutorial with some Background

Plane and axisymmetric models in Mentat & MARC Tutorial with some Background Eindhoven University of Technology Department of Mechanical Engineering Piet J.G. Schreurs Lambèrt C.A. van Breemen March 6,

### QUESTION BANK Composite Materials

QUESTION BANK Composite Materials 1. Define composite material. 2. What is the need for composite material? 3. Mention important characterits of composite material 4. Give examples for fiber material 5.

### Buckling Behavior of 3D Randomly Oriented CNT Reinforced Nanocomposite Plate

Buckling Behavior of 3D Randomly Oriented CNT Reinforced Nanocomposite Plate Outline Introduction Representative Volume Element (RVE) Periodic Boundary Conditions on RVE Homogenization Method Analytical

### COMPRESSION AND BENDING STIFFNESS OF FIBER-REINFORCED ELASTOMERIC BEARINGS. Abstract. Introduction

COMPRESSION AND BENDING STIFFNESS OF FIBER-REINFORCED ELASTOMERIC BEARINGS Hsiang-Chuan Tsai, National Taiwan University of Science and Technology, Taipei, Taiwan James M. Kelly, University of California,

### PLAT DAN CANGKANG (TKS 4219)

PLAT DAN CANGKANG (TKS 4219) SESI I: PLATES Dr.Eng. Achfas Zacoeb Dept. of Civil Engineering Brawijaya University INTRODUCTION Plates are straight, plane, two-dimensional structural components of which

### Materials and Structures. Indian Institute of Technology Kanpur

Introduction to Composite Materials and Structures Nachiketa Tiwari Indian Institute of Technology Kanpur Lecture 16 Behavior of Unidirectional Composites Lecture Overview Mt Material ilaxes in unidirectional

### Bone Tissue Mechanics

Bone Tissue Mechanics João Folgado Paulo R. Fernandes Instituto Superior Técnico, 2016 PART 1 and 2 Introduction The objective of this course is to study basic concepts on hard tissue mechanics. Hard tissue

### LAMINATION THEORY FOR THE STRENGTH OF FIBER COMPOSITE MATERIALS

XXII. LAMINATION THEORY FOR THE STRENGTH OF FIBER COMPOSITE MATERIALS Introduction The lamination theory for the elastic stiffness of fiber composite materials is the backbone of the entire field, it holds

### Problem Set 2 Due Thursday, October 1, & & & & # % (b) Construct a representation using five d orbitals that sit on the origin as a basis:

Problem Set 2 Due Thursday, October 1, 29 Problems from Cotton: Chapter 4: 4.6, 4.7; Chapter 6: 6.2, 6.4, 6.5 Additional problems: (1) Consider the D 3h point group and use a coordinate system wherein

### Materials and Structures. Indian Institute of Technology Kanpur

Introduction to Composite Materials and Structures Nachiketa Tiwari Indian Institute of Technology Kanpur Lecture 15 Behavior of Unidirectional Composites Lecture Overview Mt Material ilaxes in unidirectional

### Micro-meso draping modelling of non-crimp fabrics

Micro-meso draping modelling of non-crimp fabrics Oleksandr Vorobiov 1, Dr. Th. Bischoff 1, Dr. A. Tulke 1 1 FTA Forschungsgesellschaft für Textiltechnik mbh 1 Introduction Non-crimp fabrics (NCFs) are

### ANALYSIS OF STRAINS CONCEPT OF STRAIN

ANALYSIS OF STRAINS CONCEPT OF STRAIN Concept of strain : if a bar is subjected to a direct load, and hence a stress the bar will change in length. If the bar has an original length L and changes by an

### 3. BEAMS: STRAIN, STRESS, DEFLECTIONS

3. BEAMS: STRAIN, STRESS, DEFLECTIONS The beam, or flexural member, is frequently encountered in structures and machines, and its elementary stress analysis constitutes one of the more interesting facets

### Jeff Brown Hope College, Department of Engineering, 27 Graves Pl., Holland, Michigan, USA UNESCO EOLSS

MECHANICS OF MATERIALS Jeff Brown Hope College, Department of Engineering, 27 Graves Pl., Holland, Michigan, USA Keywords: Solid mechanics, stress, strain, yield strength Contents 1. Introduction 2. Stress

### USING A HOMOGENIZATION PROCEDURE FOR PREDICTION OF MATERIAL PROPERTIES AND THE IMPACT RESPONSE OF UNIDIRECTIONAL COMPOSITE

Volume II: Fatigue, Fracture and Ceramic Matrix Composites USING A HOMOGENIZATION PROCEDURE FOR PREDICTION OF MATERIAL PROPERTIES AND THE IMPACT RESPONSE OF UNIDIRECTIONAL COMPOSITE A. D. Resnyansky and

### 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

### 14. LS-DYNA Forum 2016

14. LS-DYNA Forum 2016 A Novel Approach to Model Laminated Glass R. Böhm, A. Haufe, A. Erhart DYNAmore GmbH Stuttgart 1 Content Introduction and Motivation Common approach to model laminated glass New

### EFFECT OF ANISOTROPIC YIELD CRITERION ON THE SPRINGBACK IN PLANE STRAIN PURE BENDING

EFFECT OF ANISOTROPIC YIELD CRITERION ON THE SPRINGBACK IN PLANE STRAIN PURE BENDING F.V. Grechnikov 1,, Ya.A. Erisov 1, S.E. Alexandrov 3 1 Samara National Research University, Samara, Russia Samara Scientific

### Chapter 5 Elastic Strain, Deflection, and Stability 1. Elastic Stress-Strain Relationship

Chapter 5 Elastic Strain, Deflection, and Stability Elastic Stress-Strain Relationship A stress in the x-direction causes a strain in the x-direction by σ x also causes a strain in the y-direction & z-direction

### 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

### Crystal Relaxation, Elasticity, and Lattice Dynamics

http://exciting-code.org Crystal Relaxation, Elasticity, and Lattice Dynamics Pasquale Pavone Humboldt-Universität zu Berlin http://exciting-code.org PART I: Structure Optimization Pasquale Pavone Humboldt-Universität

### Tuesday, February 11, Chapter 3. Load and Stress Analysis. Dr. Mohammad Suliman Abuhaiba, PE

1 Chapter 3 Load and Stress Analysis 2 Chapter Outline Equilibrium & Free-Body Diagrams Shear Force and Bending Moments in Beams Singularity Functions Stress Cartesian Stress Components Mohr s Circle for

### COURSE STE6289 Modern Materials and Computations (Moderne materialer og beregninger 7.5 stp.)

Narvik University College (Høgskolen i Narvik) EXAMINATION TASK COURSE STE6289 Modern Materials and Computations (Moderne materialer og beregninger 7.5 stp.) CLASS: Master students in Engineering Design

### Stress Integration for the Drucker-Prager Material Model Without Hardening Using the Incremental Plasticity Theory

Journal of the Serbian Society for Computational Mechanics / Vol. / No., 008 / pp. 80-89 UDC: 59.74:004.0 Stress Integration for the Drucker-rager Material Model Without Hardening Using the Incremental

### Properties of Matrices and Operations on Matrices

Properties of Matrices and Operations on Matrices A common data structure for statistical analysis is a rectangular array or matris. Rows represent individual observational units, or just observations,

### EMA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 2 Stress & Strain - Axial Loading

MA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 2 Stress & Strain - Axial Loading MA 3702 Mechanics & Materials Science Zhe Cheng (2018) 2 Stress & Strain - Axial Loading Statics

### 6. Bending CHAPTER OBJECTIVES

CHAPTER OBJECTIVES Determine stress in members caused by bending Discuss how to establish shear and moment diagrams for a beam or shaft Determine largest shear and moment in a member, and specify where

### Reflection of SV- Waves from the Free Surface of a. Magneto-Thermoelastic Isotropic Elastic. Half-Space under Initial Stress

Mathematica Aeterna, Vol. 4, 4, no. 8, 877-93 Reflection of SV- Waves from the Free Surface of a Magneto-Thermoelastic Isotropic Elastic Half-Space under Initial Stress Rajneesh Kakar Faculty of Engineering

### Anisotropic modeling of short fibers reinforced thermoplastics materials with LS-DYNA

Anisotropic modeling of short fibers reinforced thermoplastics materials with LS-DYNA Alexandre Hatt 1 1 Faurecia Automotive Seating, Simplified Limited Liability Company 1 Abstract / Summary Polymer thermoplastics

### Stacking sequences for Extensionally Isotropic, Fully Isotropic and Quasi-Homogeneous Orthotropic Laminates.

49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials 7-10 April 2008, Schaumburg, IL AIAA 2008-1940 Stacking sequences for Extensionally Isotropic, Fully Isotropic and Quasi-Homogeneous

### Transduction Based on Changes in the Energy Stored in an Electrical Field

Lecture 7-1 Transduction Based on Changes in the Energy Stored in an Electrical Field - Electrostriction The electrostrictive effect is a quadratic dependence of strain or stress on the polarization P

### 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,

### LAMINATED COMPOSITE PLATES

LAMINATED COMPOSITE PLATES David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge, MA 02139 February 10, 2000 Introduction This document is intended

The Conjugate Gradient Method Jason E. Hicken Aerospace Design Lab Department of Aeronautics & Astronautics Stanford University 14 July 2011 Lecture Objectives describe when CG can be used to solve Ax

### Solid State Physics 1. Vincent Casey

Solid State Physics 1 Vincent Casey Autumn 2017 Contents 1 Crystal Mechanics 1 1.1 Stress and Strain Tensors...................... 2 1.1.1 Physical Meaning...................... 6 1.1.2 Simplification

### Long-term Life Prediction of CFRP Structures Based on MMF/ATM Method

The 5 th Coposites Durability Workshop (CDW-5, October 7 to 20, 200, Kanazawa Institute o Technology Long-ter Lie Prediction o CFRP Structures Based on MMF/ATM Method by Yasushi Miyano and Masayuki Nakada