# Measures of Stress and Strain

• Anthony Bedford
• Kenneth M. Liechti
Chapter

## Abstract

Stresses and strains are measures used to describe the internal forces and the resulting deformations within materials. The internal force exerted on a plane within a sample of material is described by a vector-valued distributed force, the traction. The component of the traction normal to the plane is the normal stress σ, and the component tangential to the plane is the shear stress τ. The force exerted on an element of area dA is σ dA in the normal direction and τ dA in the tangential direction. The average normal and tangential forces on an arbitrary plane can be determined by integrating with respect to the area. When the external forces acting on an object in equilibrium are known, the average normal and shear stresses acting on an arbitrary plane through the object can be determined by applying the equilibrium equations. Examples include the average normal stress on a plane perpendicular to the axis of an axially loaded bar and the average shear stress on a plane perpendicular to the axis of the pin in a pin support. Given a cartesian coordinate system, consider a plane through a particular point of an object that is perpendicular to the x axis. The normal stress at the point is denoted by σx, and the shear stresses in the y and z directions are denoted by τxy and τxz, respectively. Introducing corresponding definitions using planes through the point which are perpendicular to the y and z axes results in six components of stress: σx, σy, σz, τxy, τxz, and τyz. (The shear stresses τxy = τyx, τyz = τzy, and τxz = τzx.) These components are referred to as the state of strain at the point in terms of the given coordinate system. Now consider an infinitesimal line at a given point within a material. If the material is deformed relative to its initial configuration, called the reference state, the resulting change in length of the line divided by its original length is called the normal strain in the direction of the line in the reference state, denoted by ε. If the normal strain in the direction tangential to a finite line is known at each point of the finite line, the change in its length due to the deformation can be determined by integration. If the normal strain parallel to the axis of a bar of length L is uniform throughout the bar’s length, the change in length of the bar is δ = εL. Consider two perpendicular infinitesimal lines, forming an L shape, at a given point in a reference state of a material. The decrease in the angle between the two lines resulting from a deformation, expressed in radians and denoted by γ, is called the shear strain related to the directions of the infinitesimal lines. Given a cartesian coordinate system at a particular point of a material, the normal strain in the direction parallel to the x axis resulting from a deformation is denoted by εx. The shear strain related to the positive x and y axis directions is denoted by γxy. Proceeding in this way, six strain components εx, εy, εz, γxy, γxz, and γyz are defined and referred to as the state of strain at the point in terms of the given coordinate system. For the model of material behavior called linear elasticity, the components of the state of strain at a point can be expressed as linear functions of the components of the state of stress at that point, referred to as stress-strain relations. The constants in these relations depend on the material.

## Keywords

Average stresses Elasticity Modulus of elasticity Normal strain Normal stress Poisson’s ratio Reference state Shear modulus Shear strain Shear stress State of strain State of stress Strain Strain components Stress Stress components Stress-strain relations Tractions

## Supplementary material

© Springer Nature Switzerland AG 2020

## Authors and Affiliations

• Anthony Bedford
• 1
• Kenneth M. Liechti
• 1
1. 1.University of TexasAustinUSA