Continuum Mechanics (MTH487) Lecture 7 Instructor Dr. Junaid Anjum

Aims and Objectives Body and Surface forces Mass density Stress Cauchy Stress principle The Stress Tensor Cauchy Stress formula

Stress Principles… Body forces : forces acting on all volume element, and are distributed throughout the body. Gravity and Inertia are the best know examples. We designate body forces by the vector symbol “bi” (force per unit mass) or by the symbol “pi” (force per unit volume). Surface forces : forces that act upon, and are distributed in some fashion over a surface element of the body, regardless of whether that element is part of the bounding surface or an arbitrary element of surface within the body. These are denoted by the vector symbol “fi” and have dimensions of force per unit area.

Stress Principles… Mass density: Consider a material body B having a volume V enclosed by a surface S, and occupying a regular region Ro of the physical space. Let P be an interior point of the body located in the small element of volume whose mass is as indicated in the figure below. Recall that mass is that property of a material body by virtue of which the body possesses inertia, that is, the opposition which the body offers to any change in its motion. We define the average density of this volume element by the ratio and the density at point P by the limit of this ratio as the volume shrinks to the point P.

Stress Principles… Mass density The units of density are kilograms per cubic meter (kg/m3). Notice that the two measures of body forces, bi, having units of Newton per kilogram (N/kg), and pi, having units of Newton per meter cubed (N/m3) are related through the density by the equation In general the density is a scalar function of position and time and thus may vary from point to point within the body.

Stress Principles… Cauchy Stress principle Consider a homogenous, isotropic material body B. Let P be an interior point of B and imagine a plane surface S* passing through point P (sometimes referred to as cutting plane) so as to partition the body into two portions I & II (Fig. 1) (Fig. 1) (Fig. 2)

Stress Principles… Cauchy Stress principle Consider a homogenous, isotropic material body B. Let P be an interior point of B and imagine a plane surface S* passing through point P (sometimes referred to as cutting plane) so as to partition the body into two portions I & II (Fig. 1) The point P is in the small element of area of the cutting plane which is defined by the unit normal vector pointing in the direction from portion I to portion II as shown in Fig 2. The internal forces being transmitted across the cutting plane due to the action of plane II onto plane I result in a net force and net moment . (Fig. 2) The Cauchy principle asserts that in the limiting sense i.e.

Stress Principles… Cauchy Stress principle The vector is called the stress vector. In above equations, we assume that in the limit at P, the moment vector vanishes and there is no concentrated moment of couple stress as it is called. Notice that in the symbol , the appearance of normal vector is important to emphasize that corresponding to a different cutting plane, there exists a different with a different associated stress tensor. The pair defines the state of stress at that point.

Stress Principles… Cauchy Stress principle principle of linear momentum for portion I and portion II… for the whole body on surface element S*

Stress Principles… Cauchy Stress principle Portion I onto portion II

Stress Principles… The Stress Tensor Cauchy stress principle associates a stress vector with each direction at point P. In particular, if we work in rectangular Cartesian reference frames at P, there exist vectors in each direction In terms of their coordinate components, or more compactly, using the summation convention we will now work out the expression for components of stress tensor.

Stress Principles… Cauchy Stress formula Consider the equilibrium of a small portion of body in the shape of a tetrahedron

Stress Principles… Cauchy Stress formula On the tetrahedron face ABC… Consider the equilibrium of a small portion of body in the shape of a tetrahedron On the tetrahedron face ABC…

Stress Principles… Cauchy Stress formula On the tetrahedron face ABC… Consider the equilibrium of a small portion of body in the shape of a tetrahedron On the tetrahedron face ABC…

Stress Principles… Cauchy Stress formula On the tetrahedron face PBC… Consider the equilibrium of a small portion of body in the shape of a tetrahedron On the tetrahedron face PBC…

Stress Principles… Cauchy Stress formula On the tetrahedron face PAC… Consider the equilibrium of a small portion of body in the shape of a tetrahedron On the tetrahedron face PAC…

Stress Principles… Cauchy Stress formula On the tetrahedron face PAB… Consider the equilibrium of a small portion of body in the shape of a tetrahedron On the tetrahedron face PAB…

Stress Principles… Cauchy Stress formula hence …. Consider the equilibrium of a small portion of body in the shape of a tetrahedron hence ….

Stress Principles… Cauchy Stress formula and the body force … Consider the equilibrium of a small portion of body in the shape of a tetrahedron and the body force …

Stress Principles… Cauchy Stress formula Consider the equilibrium of a small portion of body in the shape of a tetrahedron If dS is the area of the base , then the area of respective faces will be the projected are

Stress Principles… Cauchy Stress formula Consider the equilibrium of a small portion of body in the shape of a tetrahedron If dS is the area of the base , then the area of respective faces will be the projected are equilibrium requires the vector sum of all forces acting on the tetrahedron to be zero, that is

Stress Principles… Cauchy Stress formula Cauchy Stress Formula

Aims and Objectives Body and Surface forces mass density Stress The Stress Tensor Cauchy Stress Principle Cauchy Stress formula

Stress Principles… Example: Let the components of the stress tensor at P be given in matrix form by in units of mega-Pascals. Determine The stress vector on the plane at P having the unit normal The stress vector on a plane at P parallel to the plane ABC shown in the sketch.

Stress Principles… Example: Let the components of the stress tensor at P be given in matrix form by in units of mega-Pascals. Determine The stress vector on the plane at P having the unit normal

Stress Principles… Example: Let the components of the stress tensor at P be given in matrix form by in units of mega-Pascals. Determine (b) The stress vector on a plane at P parallel to the plane ABC shown in the sketch.

Stress Principles… Example: Let the components of the stress tensor at P be given in matrix form by in units of mega-Pascals. Determine (b) The stress vector on a plane at P parallel to the plane ABC shown in the sketch.