# State of Stress at Point

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1 State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention, when an index variable appears twice in a single term, once in an upper (superscript) and once in a lower (subscript) position, it implies that we are summing over all of its possible values.

2 Vector Representation in Einstein Indicial Notation x 3 e 1 e 3 θ 1 θ 3 a θ 2 n e 2 x 1 x 2 Unit Vector =

3 Vector dot product In mechanics and engineering, vectors in 3D space are often described in relation to orthogonal unit vectors i, j and k. If the basis vectors i, j, and k are instead expressed as e 1, e 2, and e 3, a vector can be expressed in terms of a summation: In Einstein notation, the summation symbol is omitted since the index i is repeated once as an upper index and once as a lower index, and we simply write Using e 1, e 2, and e 3 instead of i, j, and k, together with Einstein notation, we obtain a concise algebraic presentation of vector and tensor equations. For example,

4 Kronecker's delta function The Kronecker delta function or Kronecker's delta function, is a function of two variables, usually integers, which is 1 if they are equal and 0 otherwise. So, for example, δ 1,2 = 0, but δ 3,3 = 1 It is written as the symbol δ ij, and treated as a notational shorthand rather than as a function.

5 Abstract definitions and Examples In the traditional usage, one has in mind a vector space V with finite dimension n, and a specific basis of V. We can write the basis vectors as e 1, e 2,..., e n. Then if v is a vector in V, it has coordinates relative to this basis. The basic rule is: In this expression, it was assumed that the term on the right side was to be summed as i goes from 1 to n An index that is summed over is a summation index. Here, the i is known as a summation index. It is also known as a dummy index since the result is not dependent on it; thus we could also write, for example: Matrix multiplication We can represent matrix multiplication as: Examples: 4-Dimensional space indices run from 0 to 3 : Vector cross product: where with ε ijk

6 State of stress at a point. Stress is a measure of the internal forces acting within a deformable body. Quantitatively, Stress is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body. Because the loaded deformable body is assumed to behave as a continuum, these internal forces are distributed continuously within the volume of the material body, and result in deformation of the body's shape. Beyond certain limits of material strength, this can lead to a permanent shape change or physical failure. Stress in a loaded deformable material body assumed as a continuum Axial stress in a prismatic bar axially loaded

7 Defintions Continuum mechanics deals with deformable bodies The stresses considered in continuum mechanics are only those produced by deformation of the body by surface forces contact forces, can act either on the bounding surface of the body and Body Forces originate from sources outside of the body that act on its volume (or mass) (gravitational field). When external contact forces act on a body, internal contact forces pass from point to point inside the body to balance their action, according to Newton's second law of motion of conservation of linear and angular momentum (principle of transmissibility). These laws are called Euler's equations of motion for continuous bodies. The density of internal forces at every point in a deformable body is not necessarily even, i.e. there is a distribution of stresses. This variation of internal forces is governed by the laws of conservation of linear and angular momentum, which normally apply to a mass particle but extend in continuum mechanics to a body of continuously distributed mass. Let us consider such a body subjected to contact and body forces as shown in this Figure. The Cauchy Stress Principle provides a framework for calculating The state of stress at any point P of the body under arbitrary distributions of contact forces F i and Body forces b i :

8 Cauchy stress principle The Cauchy stress principle states that upon any surface (real or imaginary) that divides the body, the action of one part of the body on the other is equivalent (equipollent) to the system of distributed forces and couples on the surface dividing the body, and it is represented by a vector field T (n), called the stress vector, defined on the surface S and assumed to depend continuously on the surface's unit vector n. [9] Internal distribution of contact forces and couple stresses on a differential of the internal surface in a continuum, as a result of the interaction between the two portions of the continuum separated by the surface F i Internal distribution of contact forces and couple stresses on a differential of the internal surface in a continuum, as a result of the interaction between the two portions of the continuum separated by the surface

9 Traction Vector or Stress Vector T ( n) consider an imaginary surface S passing through an internal material point P dividing the continuous body into two segments. The forces transmitted from one point to another generate a distribution on a small surface area ΔS, with a normal unit vector n, on the dividing plane S. Cauchy s stress principle asserts that as ΔS becomes very small and tends to zero the ratio ΔF/ΔS becomes df/ds and the couple stress vector ΔM vanishes. The resultant vector df/ds is defined as the stress vector or traction vector given by T (n) = T i (n) e i at the point P associated with a plane with a normal vector n: Cauchy s fundamental lemma is equivalent to Newton's third law of motion of action and reaction, and is expressed as The State of Stress at Point P is defined as the set : { T (n) n through P }

10 Normal and Shear Stress One normal to the plane, called normal stress with where df n is the normal component of the force df to the differential area ds and the other to this plane, called the shear stress where df s is the tangential component of the force df to the differential surface area ds Stress vector on an internal surface S with normal vector n. Depending on the orientation of the plane under consideration, the stress vector may not necessarily be perpendicular to that plane, i.e. parallel to, and can be resolved into two components: one component normal to the plane, called normal stress, and another component parallel to this plane, called the shearing stress.

11 The Stress Tensor The State of Stress at Point P is defined as the set :{ T (n) n through P } This requires a knowlwdge of infinite number of T (n) s. Cauchy theorem reduces this set to merely by knowing the stress vectors on three mutually perpendicular planes, the stress vector on any other plane passing through that point can be found through coordinate transformation equations To prove this theorem, consider a tetrahedron with three faces oriented in the coordinate planes, and with an infinitesimal area da oriented in an arbitrary direction specified by a normal vector n in Figure. Note that unit vector perpendicular to x 1 Ox 3 is e 3 stress vector is -T (e 3 ), The tetrahedron is formed by slicing the infinitesimal element along an arbitrary plane n. The stress vector on this plane is denoted by T (n). The stress vectors acting on the faces of the tetrahedron are denoted as T (e 1 ), T (e 2 ), and T (e 3 ) Stress vector acting on a plane with normal vector n.

12 The Stress Tensor where the right-hand-side represents the product of the mass enclosed by the tetrahedron and its acceleration: ρ is the density, a is the acceleration, and h is the height of the tetrahedron, considering the plane n as the base The area of the faces of the tetrahedron perpendicular to the axes can be found by projecting da into each face (using the dot product): A note on the sign convention: The tetrahedron is formed by slicing a parallelepiped along an arbitrary plane n. So, the force acting on the plane n is the reaction exerted by the other half of the parallelepiped and has an opposite sign. h 0

13 The Stress Tensor We define the stress tensor component as the jth component of the stress vector The nine components σ ij of the stress vectors are the components of a second-order Cartesian tensor called the Cauchy stress tensor, which completely defines the state of stress at a point and is given by where σ 11, σ 22, and σ 33 are normal stresses, and σ 12, σ 13, σ 21, σ 23, σ 31, and σ 32 are shear stresses. The first index i indicates that the stress acts on a plane normal to the x i -axis, and the second index j denotes the direction in which the stress acts.

14 The Stress Tensor Using the components of the stress tensor Or Examples of components of the stress tensor Shear stress component on Normal stress component plane plane σ 11, σ 22, and σ 33 are normal stresses, and σ 12, σ 13, σ 21, σ 23, σ 31, and σ 32 are shear stresses. Components of stress in three dimensions

15 Transformation of Stress Tensor where A is a rotation matrix with components a ij. In matrix form this is The other components of the expanded stress tensor can be obtained by cyclic permutation The stress tensor is a second order tensor, which is a statement of how it transforms under a change of the coordinate system. From an x i - system to an x i '-system, the components σ ij in the initial system are transformed into the components σ ij ' in the new system according to :

16 Normal and Shear Stresses The magnitude of the normal stress component σ n of any stress vector T (n) acting on an arbitrary plane with normal vector n at a given point, in terms of the components σ ij of the stress tensor σ, is the dot product of the stress vector and the normal vector: The magnitude of the shear stress component τ n, acting in the plane spanned by the two vectors T (n) and n, can then be found using the Pythagorean theorem where

17 Principal stresses and stress invariants At every point in a stressed body there are at least three planes, called principal planes, with normal vectors, called principal directions, where the corresponding stress vector i perpendicular to the plane, i.e., parallel or in the same direction as the normal vector, and where there are no normal shear stresses. The three stresses normal to these principal planes are called principal stresses. A stress vector parallel to the normal vector is given by: where is a constant of proportionality, and in this particular case corresponds to the magnitudes of the normal stress vectors or principal stresses. Knowing that This is a homogeneous system, i.e. equal to zero, of three linear equations where are the unknowns. To obtain a nontrivial (non-zero) solution for, the determinant matrix of the coefficients must be equal to zero, i.e. the system is singular. Thus

18 Principal stresses and stress invariants Expanding the determinant leads to the characteristic equation where The characteristic equation has three real roots, i.e. not imaginary due to the symmetry of the stress tensor. The three roots,, and are the eigenvalues or principal stresses, and they are the roots. The principal stresses are unique for a given stress tensor. Therefore, from the characteristic equation, the coefficients, and, called the first, second, and third stress invariants, respectively, have always the same value regardless of the coordinate system's orientation.

19 For each eigenvalue, there is a non-trivial solution for in the equation. These solutions are the principal directions or eigenvectors defining the plane where the principal stresses act. The principal stresses and principal directions characterize the stress at a point and are independent of the orientation. A coordinate system with axes oriented to the principal directions implies that the normal stresses are the principal stresses and the stress tensor is represented by a diagonal matrix: For which the three stress Invariants are: Maximum and minimum shear stresses The maximum shear stress or maximum principal shear stress is equal to one-half the difference between the largest and smallest principal stresses, and acts on the plane that bisects the angle between the directions of the largest and smallest principal stresses, i.e. the plane of the maximum shear stress is oriented from the principal stress planes. The maximum shear stress is expressed as assming

20 Stress deviator tensor The stress tensor can be expressed as the sum of two other stress tensors: a mean hydrostatic stress tensor or volumetric stress tensor or mean normal stress tensor,, which tends to change the volume of the stressed body; and a deviatoric component called the stress deviator tensor, which tends to distort it where is the mean stress given by: Note that convention in solid mechanics differs slightly from what is listed above. In solid mechanics, pressure is generally defined as negative one-third the trace of the stress tensor. The deviatoric stress tensor can be obtained by subtracting the hydrostatic stress tensor from the stress tensor:

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28 Stress deviator tensor Stress deviator tensor Stress deviator tensor Invariants

29 PROBLEMS SOLVED ON STRESS TENSOR 1. 2.

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