Strain is also a symmetric secondorder tensor, identical to the stress. Kolecki national aeronautics and space administration glenn research center cleveland, ohio 445 tensor analysis is the type of subject that can make even the best of students shudder. Further simplification based on experiments observations. Each diagonal component of the strain deviator tensor. Its predictive capability is better than shear lag models and the errors associated with applying it to nonellipsoidal geometries cylinders, plates, etc. Stress strain relationship, hookes law, poissons ratio, shear stress, shear strain, modulus of rigidity.
Many physical properties of crystalline materials are direction dependent because. Here, the stress is defined more formally, cauchys law cauchys law states that there exists a cauchy stress tensor. Our focus is on relating deformation to quantities that can be measured in the. Quantify the linear elastic stress strain response in terms of tensorial quantities and in particular the fourthorder elasticity or sti ness tensor describing hookes law. Stress analysis by using strain gages introduction a state of strain in 3d the 3d strain tensor may be characterized by its.
In the approximation of the elastic continuum, the position of each point of a solid is described by the vector which in some cartesian coordinate system has the components. Chapter 3 the stress tensor for a fluid and the navier. Deformation, stress, and conservation laws in this chapter, we will develop a mathematical description of deformation. For example, the tensor relating stress and strain tensors in the stress strain relations is of fourth order and is referred as such.
Strain can also be rotated to find its principal strain, principal strain direction, and maximum shear strain. These six scalars arranged in an ordered matrix forms the stress tensor. To introduce the deformation tensor, we first consider the deformation of an infinitesimal material. Opening remarks this document was written for the bene.
Conventionally, a shear strain is defined by the shear angle produced in simple shear, below. Under strain all points of a solid are in general shifted. For an infinitesimal fibre that deforms from an initial point given by the vector dx to the vector dx in the time t, the deformation gradient is defined by fij. An introduction to tensors for students of physics and. Therefore, there are 6 independent variables in the strain matrix, instead of 9. The values of these six components at the given point will change with.
A number of forces are acting on this body in different directions but the net force the vector sum of the forces on the body is 0. The relationship of stress and strain for steel can be observed in. For example, if you calculate the principal values of a cauchy stress tensor, then what you get are. A tensor is an object that speci es some physical or abstract condition e. Ax fl e a graph of stress against strain will be a straight line with a gradient of e. Since the stress and strain at a point are tensor quantities, a simple definition of tensors involving transformation of components with respect to two cartesian. For large deformations, also called finite deformations, other measures of stress are required, such as the piolakirchhoff stress tensor, the biot stress tensor, and the. The stress at a point of an elastic solid is an example of a tensor which depends on two directions one normal to the area and other that of the force on it. Principal stresses and strains continuum mechanics. Mechanics of solids mechanics of solids finite deformation and strain tensors. Normal stresses are given by the diagonal elements. Imagine an arbitrary solid body oriented in a cartesian coordinate system.
Definition of stress, stress tensor, normal and shear stresses in axially loaded members. Page 1 15 me 410 mechanical engineering laboratory experiment 5. The cauchy stress tensor is used for stress analysis of material bodies experiencing small deformations. Molecular spectroscopy workbench stress, strain, and raman. After showing its tensor character, all results we obtained for stress tensors can be translated to strains. The stress tensor for a fluid and the navier stokes equations 3. Basic concepts of stress and strain mechanics of materials describes how solid materials will deform change shape and how they will fail break when subjected to.
The elements of the stress tensor have units of pressurenamely, force per unit area. Module 3 constitutive equations learning objectives understand basic stress strain response of engineering materials. Chapter 2 deformation and strain lecture 5 strain tensors content. Although we have not yet discussed the many different definitions of stress and strain, it is in fact true that everything discussed here applies regardless of the type of stress or strain tensor. The infinitesimal strain theory is commonly adopted in civil and mechanical engineering for the stress analysis of structures built from relatively stiff elastic materials like concrete and steel, since a common goal in the design of such structures is to minimize their deformation under typical loads.
Elements of continuum elasticity mit opencourseware. Because of these symmetries, the stress tensor is a symmetric tensor, and thus has only 6 independent elements. The stress tensor the general state of stress in any homogeneous continuum, whether. To get unique relations for the strain tensor, compatibility. The fundamental difference between strain and stress is that strain terms reflect descriptive interpretations of what movements produced a structure, while stress terms reflect genetic. It is a central concept in the linear theory of elasticity. The governing equations and some intermediate equations are expressed in tensor notation. In the theory of finite deformations, extension and rotations of line elements are unrestricted as to size. An introduction to tensors for students of physics and engineering joseph c.470 973 456 567 51 581 1095 619 923 1336 935 1356 1573 466 959 733 250 229 815 631 417 228 591 43 734 1353 1377 7 1084 1248