Mae 640 Lec11

Continuum mechanics – MAE 640 Summer II – 2009

Dr. Konstantinos Sierros 263 ESB new add [email protected]

Chapter 4: Stress Measures Introduction •All materials have certain threshold to withstand forces, beyond which they “fail” to perform their intended function. •The force per unit area, called stress, is a measure of the capacity of the material to carry loads. •It is necessary to determine the state of stress in a material. •In this chapter we study the concept of stress and its various measures. For example, stress at a point in a three-dimensional continuum can be measured in terms of nine quantities, three per plane, on three mutually perpendicular planes at the point. These nine quantities may be viewed as the components of a second-order tensor, called stress tensor.

Cauchy Stress Tensor and Cauchy’s Formula •First we introduce the true stress which is the stress in the deformed configuration κ that is measured per unit area of the deformed configuration κ. •The surface force acting on a small element of area in a continuous medium depends not only on the magnitude of the area but also upon the orientation of the area. •It is common to denote the direction of a plane area by means of a unit vector drawn normal to that plane, as discussed in Section 2.2.3.

Cauchy Stress Tensor and Cauchy’s Formula

•Let the unit normal vector be denoted by ˆn. Then the area is expressed as A = Aˆn. •If we denote by df( ˆn) the force on a small area ˆnda located at the position x, the stress vector can be defined.

Cauchy Stress Tensor and Cauchy’s Formula

stress vector

Cauchy Stress Tensor and Cauchy’s Formula

•The stress vector is a point function of the unit normal ˆn which denotes the orientation of the surface a. •The component of t that is in the direction of ˆn is called the normal stress. •The component of t that is normal to ˆn is called the shear stress.

Cauchy Stress Tensor and Cauchy’s Formula

cause of Newton’s third law for action and reaction, we have;

•At a fixed point x for each given unit vector ˆn, there is a stress vector t( ˆn) acting on the plane normal to ˆn. •Note that t( ˆn) is, in general, not in the direction of ˆn. It is good to establish a relationship between t and ˆn.

Cauchy Stress Tensor and Cauchy’s Formula •To establish the relationship between t and ˆn, we now set up an infinitesimal tetrahedron in Cartesian coordinates, as shown in the figure below;

−t1,−t2,−t3, and t denote the stress vectors in the outward directions on the faces Areas of the infinitesimal tetrahedron

Cauchy Stress Tensor and Cauchy’s Formula •Using Newton’s second law for the mass inside the tetrahedron, acceleration

body force per unit mass

density

volume of the tetrahedro

•Since the total vector area of a closed surface is zero (using the gradient theorem), we have;

The volume of the element v can be expressed as;

perpendicular distance from the origin to the slant face.

Cauchy Stress Tensor and Cauchy’s Formula

…and using these expressions;

And dividing by Δα we have;

…when the tetrahedron shrinks to a point (i.e.Δh→ 0);

Cauchy Stress Tensor and Cauchy’s Formula

using the summation convention

stress dyadic or stress tensor σ •The stress tensor is a property of the medium that is independent of the ˆn. Therefore we have;

Cauchy Stress Tensor and Cauchy’s Formula

The stress vector t represents the vectorial stress on a plane whose normal is ˆn

Cauchy stress formula Cauchy stress tensor. the current force per unit deformed area,

Cauchy Stress Tensor and Cauchy’s Formula •In Cartesian component form, the Cauchy formula

Can be written as; ti = njσji …and in matrix form…

stress (force per unit area) on a plane perpendicular to the xi coordinate and in the xj coordinate direction

Cauchy Stress Tensor and Cauchy’s Formula The component σi j represents the stress (force per unit area) on a plane perpendicular to the xi coordinate and in the xj coordinate direction

Transformation of Stress Components •The Cauchy stress is a second-order tensor therefore we can define; •Its invariants, •Transformation law •Eigenvalues and eigenvectors. Invariants

Transformation law

Principal stresses and principal planes Eigenvalues and eigenvectors •The determination of maximum normal stresses and shear stresses at a point is of considerable interest in the design of structures because failures occur when the the magnitudes of stresses exceed the allowable (normal or shear) stress values, called strengths, of the material. •It is of interest to determine the values and the planes on which the stresses are the maximum •Therefore, we must determine the eigenvalues and eigenvectors associated with the stress tensor

Yielding a cubic equation for λ, called the characteristic equation, the solution of which yields three values of λ. •The eigenvalues λ of σ are called the principal stresses and the associated eigenvectors are called the principal planes.