Difference between first and second piolakirchhoff stress vector. Tensors have their applications to riemannian geometry, mechanics, elasticity, theory of relativity. Our focus is on relating deformation to quantities that can be measured in the. If you pick three perpendicular surfaces of these cubes you can def. Deformation, stress, and conservation laws in this chapter, we will develop a mathematical description of deformation. The cauchy stress tensor for a material subject to an isotropic internal constraint article pdf available in journal of engineering mathematics 371. Piolakirchhoff stress tensor, in continuum mechanics. The symmetry of the stress tensor a polar fluid is one that is capable of transmitting stress couples and being subject to body torques, as in magnetic fluids.
In continuum mechanics, the cauchy stress tensor, true stress tensor, or simply called the stress tensor is a second order tensor named after augustinlouis cauchy. Math 575lecture 2 1 conservation of momentum and cauchy. Stress, cauchys equation and the navierstokes equations 3. The stress tensor the secondorder tensor which we will be examining has. I imagine you have already looked into the wikipedia stuff so ill just explain how i understand it personally. Using the gradient operator with respect to the undeformed state, the deformation tensor can be written as a gradient. Sign convention the cauchy formula can also be consistently used to determine the sign of the components of the stress tensor.
Since both the cauchy stress tensor and the small strain tensor are defined in the deformed coordinate system, they are energetically conjugate. However, the deformed configuration of a body is usually unknown and it is usually what we aim to. This is cauchy s equation of motion and a is the acceleration. If a function f is analytic on a simply connected domain d and c is a simple closed contour lying in d then. Cauchys formula expresses tn as a function of the stress vectors on the planes perpendicular to the coordinate axes, i. The cauchy momentum equation is a vector partial differential equation put forth by cauchy that describes the nonrelativistic momentum transport in any continuum. The tensor relates a unitlength direction vector n to the traction. Chapter 3 the stress tensor for a fluid and the navier stokes equations 3. To this end we consider a small tetrahedron of volume v centred in p. The tensor c ftf the right cauchygreen deformation tensor is thus a measure of the change in ds2 0 due to gradients of the motion ds2.
The symmetry of the cauchy stress tensor is the result of applying conservation of angular momentum to an infinitesimal material element. Quantify the linear elastic stressstrain response in terms of tensorial quantities and in particular the fourthorder elasticity or sti ness tensor describing hookes law. Consider an elementary surface, da, with unit normaln, through a typical pointx in the deformed con. The symmetry of the stress tensor a polar fluid is one that is capable of transmitting stress couples and being subject to.
Electromagnetic stressenergy tensor, in relativistic physics. The principle is needed to explain where the cauchy tensor comes from. We can extend the cauchy stress tensors application to an arbitrary direction by considering a tetrahedron as situated in figure 2b. Difference between first and second piolakirchhoff stress. This tensor yields the local stress on a surface by. Stress components taking cauchy s law to be true it is proved below, the components of the stress tensor with respect to a cartesian coordinate system are, from 1.
Cauchygreen tensor an overview sciencedirect topics. This derivation assumes that there are no body moments. However, for the finite strain tensor, since it is defined in the reference coordinate system, we need to derive a new stress tensor. Stress components taking cauchys law to be true it is proved below, the components of the stress tensor with respect to a cartesian coordinate system are, from 1. Chapter 5 then addresses constitutive equations for elastic materials and viscous fluids, which are presented as tensor equations relating the tensor concept of stress to the tensors describing deformation, rate of deformation and rotation. However, in the linear elastic case, we have a linear proportion between the cauchy stress tensor and the small deformation strain tensor. Stress balance principles 03 the cauchy stress tensor.
For the love of physics walter lewin may 16, 2011 duration. Cauchys development of the stress tensor is discussed along with cauchys spatial equations of motion. The tensor c can be equivalently substituted by the tensor c. The first piola kirchoff stress tensor relates the cauchy stress tensor to the stress in the deformed space. Cauchy stress tensor an overview sciencedirect topics. The tensor consists of nine components that completely define the state of stress at a point inside a material in the deformed state, placement, or configuration. The stress vector t o, defined by the preceding equation, is a pseudostress vector in that, being based on the undeformed area, it does not describe the actual intensity of the force df, which acts on the deformed area da dan.
The cauchy stress tensor and eulercaucy stress priciple articles are better placed together. Stress in uids the stress tensor the stress tensori cauchys stress principle we now wish to characterise the state of stress at a point p of a continuum. Considering a general anisotropic hyperelastic solid, with the strain energy. Components of the stress tensor on three facets of the in nitesimal surface element. Examples are given of several special stress fields possessing particular structures, and the stress power is introduced. We note that t o has the same direction as the cauchy stress vector t the first piolakirchhoff stress tensor also known as the lagrangian stress tensor is a. It holds for any continuum no matter how the stress tensor t is connected with the rate of strain. The final stress state consists entirely of a normal ycomponent because the ydirection is the direction of the tensile force. P jf 1 s sft 1 s jf 1 s f t pf t 2 where j is kfk, and t denotes a transpose. If there are body moments, the only way that angular momentum can. The stress tensor is a representation of stress on three mutually perpendicular planes in a coordination system. 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.
Cauchys equation of motion the lhs therefore becomes. Stress, cauchys equation and the navierstokes equations. The cauchy stress principle states that as the area around p shrinks to zero, the following limit holds. Since we were operating in the realm of small deformation mechanics, then the cauchy stress tensor is equivalent to the other stress tensors. Cauchy stress tensor field is defined over the deformed configuration of a structure body and is directly related to the governing equations of equilibrium and boundary conditions. Think of dividing some continuous material into super small cubes. Module 3 constitutive equations learning objectives understand basic stressstrain response of engineering materials.
Chapter 4 discusses a new family of tensors related to the deformation of continuous material. A cauchys formula b principal stresses eigenvectors and eigenvalues ii cauchys formula a relates traction vector components to stress tensor components see figures 5. The corresponding referential equations of motion are obtained in terms of the piolakirchhoff stress tensor. Both mathematical and engineering mi stakes are easily made if this crucial difference is not recognized and understood. This is cauchys equation of motion and a is the acceleration. Also, defining new stress tensors is also important when we look at. Chapter 3 the stress tensor for a fluid and the navier. The point is that the sign of the components of the vectors is known from the chosen coordinate system. For example, if there is only gravitational force, then f v z v. On the nature of the cauchy stress tensor the cauchy stress tensor. Pdf the cauchy stress tensor for a material subject to. Cauchys proof for the existence of stress tensor is called cauchy tetrahedron argument. In the gure on the right e i denotes the unit vector in the direction of the axis x i i 1.
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