Lec 3 : Stress acting at a point- Stress tensor

 So, welcome back, this is the next lecture on stress tensors. So in the last lecture, we have categorically seen what is Cauchy’s stress, sigma.   And we have seen that the definition of sigma indicates what is the internal force that gets developed within a plane or a body at a point due to some action of external forces.   Now, what is sigma? In fact sigma is a stress tensor. So, Cauchy stress can be considered a tensor. So, now we are going to define a new term what is known as a tensor. If you want to study or if you want to do modeling in continuum mechanics,   as the complexity of the problem increases, it is always convenient to define what is known as a tensor. And we have already stated stress is a tensor quantity. Now, what is a tensor?   We know what are scalars, and we know what are vectors. So tensor is also a similar kind of quantity. So why tensor, because it is very convenient to express stress as a tensor.  In short stress itself is a tensor. In simple terms, we can say that tensor can be defined as a quantity with magnitude and multiple spatial directions.   So, possibly you will think like what is the difference between a tensor and a vector.   Vector also has a magnitude and a direction, but we will see that vector has magnitude,   but it will have only one direction whereas in the case of tensor multiple directions are there.  So that is the essential difference and tensor is a more general term.   And the subsets of the tensor are scalar, vector, and any other tensor of higher order. So different tensors which are popularly used are yes, that is what I told the first one is scalar the simplest tensor is scalar and it is called zero-order tensor. A quantity which has only magnitude and zero direction, scalar we all of us know that it does not have any direction it has only magnitude.  So we call it zero direction. And zero order tensor which is a scalar that has three raised to zero, where zero represents the number of directions. So three raises to zero are equal to one element and that is true, it is merely a number that shows the magnitude. Scalar is a zero-order tensor. The second one is a vector, which is the first-order tensor. A vector is a quantity,   which has magnitude and one direction you can see that vector has only one direction.  Accordingly, the number of elements will be three raised to one which is equal to three elements.   So if you have an x y z axis you have a vector in three different directions. So that is possible.   So that is what it means it has three elements. So it has one direction every vector is associated with only one direction. And it has three raised to one which is three elements and specifically, Cauchy stress is known as a second order tensor.  Why does Cauchy stress sigma have magnitude and two directions? Now, what are these two directions?   Now we will see specifically how these two directions come into the picture when you define a stress component and it is very easy also if you remember Cauchy stress, we represent it as sigma xx or sigma xy. So there are two symbols associated and that is why it is always associated with two directions it is associated with which plane it acts, which means the normal to that particular plane. It is also dependent on, which direction that particular traction acts. So we will discuss that a bit later, only to specify here is Cauchy stress is a second-order tensor. It has magnitude and it has two directions.   So this is a second-order tensor and it has three squares equal to nine elements which we have already seen in the Cauchy stress tensor. There are nine elements and the second order tensor linearly maps to vectors that also we have seen. We have seen that t is sigma transpose of n. So it linearly transforms to vectors, that is Cauchy’s formula. Now, some aspects of tensor to be very specific, may not be useful, but then this is important to understand the tensor. Let us say two, there are two vectors u and v a tensor T is a second order tensor if it linearly maps vector v to u as can be shown here and the second order tensor satisfies the properties of linear transformation. So this is what has been written, t maps v to u   or there is a linear mapping of v to u. If you compare this with Cauchy’s formula, it is more or less the same thing that is how we define The Cauchy stress tensor is a second-order tensor.   Having said that, now the next job is to interpret the components of the Cauchy stress tensor. We know that there are nine elements. Now, what are these nine elements? What does it represent? So, for that, we need to define the Cartesian coordinate. So you have a Cartesian coordinate x y   and z. And to make it simple a control volume is also shown,   control volume is a very common terminology that is used in continuum mechanics or any other form of mechanics. The con this control volume is not required, but to make things simple and for one to understand it has been shown. So we have a Cartesian coordinate x y z,   this is positive x direction, this is positive y and this is positive z which is also important here. So you can consider a positive x plane, now x is an axis which is meant by x plane. It means the plane on which x direction is the normal to that plane that is what is written here the plane whose normal is in the positive x direction. So it is called the positive x plane.  So what will be the negative x plane? The negative x plane will be here because the outward normal to this plane is in the negative x direction. So this is the negative x plane.   So you need to understand this very carefully. Consider positive x planes,   so we are talking about this particular plane. That is a positive x plane because why because normal to this particular plane x this is the y z plane, this is y, this is z. So this plane is y z plane.  Now for this y z plane, the normal is in the direction of x. So that is what it means.   So x plane means, positive x plane means, y z plane which is shown here so, positive x,   negative x plane both are there. Now we will come back to Cauchy’s formula and cauchys stress sigma. So now, the normal vector to x plane¬ this positive x direction.   Please understand the normal vector to x plane¬. So this is the x plane.  The normal is x. So you can easily write what is the normal vector. So this is the normal vector. So for the x direction, it is one zero.   So n T is given in this manner and similarly, for y it will be zero one zero, and for z zero one.   So normal vector to x plane is defined that is n transpose is given.   Now, what are the components of traction vector tx, ty, and tz? You already have this to be,   that is tx, ty, tz is equal to sigma and n. So if you substitute the value of n that is for positive x plane one zero here, so it will be one zero, and do the matrix operation,   you will see that t x is equal to sigma xx, ty will be equal to sigma xy. So this is sigma xx,   sigma xi. So ty will be equal to sigma x y and t z is equal to sigma x z. So what does it mean,   it means that the components which are present in cauchys stress tensor are components of traction vector in a given direction. So   if you see, you can see that sigma xx is the x component of traction vector on x plane. So there are two references which are coming and that is why we said that there are two directions.   It is the x component that is the traction vector in the x direction and it is acting on the x plane.   So there are two things which are coming. Similarly, you have sigma xy. Sigma XY is the y component of the traction vector acting on the x plane. Similarly, you have the z component of the traction vector acting on the x plane. Similarly, other components of Cauchy stress tensor can be identified based on   Cauchy’s formula. So that is what is the meaning of each of the terms which are present in the   Cauchy stress, it is nothing but the components of the traction vector acting in a specific direction. So, the component of the Cauchy stress tensor, in general, is sigma ij, it is the j component of the traction vector it can be x, it can be y, and it can be z. So it is a j   component of traction vector acting on the ith plane. So the first index I show which plane it is associated with. Which plane means, which is the normal and j   is the direction of that component, direction of the component of traction vector.  So i is the plane on which traction is considered, and j is the direction in which the traction component is considered.  So we can see the overall representation of the Cauchy stress tensor.   So first is stresses acting on x plane. Now which is the x plane,   this is the x plane. So there are two x planes, this is negative x and this is positive x. So what are the stresses which are acting,   we have sigma xx in the direction of x? So, all of them are acting on x plane.  Then we have sigma XY and sigma xz. Similarly, on the other side of another plane which is a negative x   plane we have sigma XY, sigma xz, and sigma xx. It is identical but it is on the other side.   Then we have stressed on y plane. Now what is meant by y plane, a plane with y direction as the normal. So you are talking about this and this. So you have positive y and this is negative y.  Similarly, in this, you have sigma yy, sigma yy, which is the direction, in the direction of y and you have sigma y x, sigma yz,   similarly sigma y x and sigma yz. Then we have stresses acting on the z plane,   what are the stresses acting on the z plane, and what is the z plane, this is the positive z plane and this is the negative z plane and this stress is acting as sigma zz,   sigma zx, sigma zy. Similarly, here also you have sigma zz, sigma zx, and sigma zy.   So these are the representation of the components of Cauchy stress on a given control volume.  So, all these stress components are acting at a point. Now we need to keep in mind that I   have shown a control volume in the figure, and that is only for understanding how the stresses are oriented. Otherwise, it does not serve any purpose. We need to still understand that whatever stress components are there in the Cauchy stress tensor, it is acting at a point and the control volume, the cuboid is shown only to indicate the plane on which it is acting.  So that’s that notion we should not forget. So it is stress acting at a point.   Now having said that, we need to now define some sort of sign convention of the Cauchy stress tensor. So the given sign conventions are the traction components on the positive plane. So now we have already marked what is a positive plane. So the traction component on the positive plane acting in the positive direction means the direction of x y z which is in the positive direction, so is positive. So you have a positive plane and the traction component is acting in the positive direction, so it is positive. Similarly, if you have a positive plane and the traction component is acting in a negative direction, so it is negative.  For the negative plane, if the plane is negative and the traction component is acting in a negative direction, so it is positive. And the final case is negative plane traction component direction is a positive direction, it is negative. So this is one sign convention, you can see that numerous sign conventions are available, and uh one may use them at his convenience, but if you follow one sign convention, you need to follow it throughout.  So this is one convenient way of uh defining sign convention, there are assigned conventions that are available based on movement also, and sometimes it may be difficult to understand. So this is very easy and very easy to define as well, one example is given here. So this is the positive x plane,   and the stresses acting are sigma xx, sigma XY and sigma xz. If you consider sigma xx,   this is acting on a positive x plane. And sigma xx is in acting in the positive x direction. So that is why it is positive, similarly to sigma XY and sigma xz. Now consider the case of the negative x plane, if you consider sigma XY, this is a negative plane, negative x plane whereas, this is acting in the positive y direction. So negative plane positive y direction,   so it is negative. Similarly, all the stress component signs can be assigned. So this is the sign convention of the Cauchy stress tensor. So what is the summary that we have understood till now? There are three normal components or normal stresses,   sigma xx, sigma yy, and sigma oz. You can see that in this figure, you have sigma x x,   sigma yy, and sigma zz, these are acting in the same direction as that normal. So there are three normal components or normal stresses sigma xx, sigma yy, sigma zz or it is merely stated as sigma x, sigma y, sigma z which is a common terminology,   which we normally use in mechanics. There are six shear components or shear stresses to be very specific all indices were I not equal to j, here it is I equal to,   here I not equal to j. So these are shear components of traction or shear stresses,   it is written either in sigma form or in tau form. Cauchy stress tensor a second-order tensor quantifies the internal force distribution in a body at a given position and time corresponding to a given deformation. Why time is important is because we are considering the condition corresponding to a given deformation.   And internal forces, which that gets developed followed the basic laws of mechanics. Now one particular aspect of why stress at a point that information is needed is to, define the equilibrium equation. So it is an application of why you need to know stress at a point. Now stress at a point is very important to define the equilibrium equation as we have seen in the beginning, you have seen that certain requirements need to be satisfied like the equilibrium condition, the compatibility condition, and so on.  Now for defining the equilibrium condition, we need to specify the equilibrium equation. I will not go into the derivation of this equilibrium equation it is very basic and is mostly seen by most of you. So by considering a given control volume, the equilibrium equation can be represented as follows. And you can see that the components of equilibrium equations are the stress tensor components, the only new term is gamma. Where gamma is a self-weight of the gravity stress which acts in the z-direction or the vertical direction and to be very specific stresses are in terms of total stresses in this particular equation, it is invariably necessary to know stress at a point for defining equilibrium condition. Now based on equilibrium,   we can say that tau yx is equal to tau XY, tau yz is equal to zy and tau zx is equal to tau xz.  Therefore, the stress tensor is represented by six independent stress components,   there are nine components in the Cauchy stress tensor just because of this condition,   we have six independent stress components, and they are three normal stresses, sigma x, sigma y,   sigma z, and three shear stresses tau XY, tau yz and tau Zx, where tau XY is equal to tau yx.   So this is what it is. So that is how it boils down to six independent stress components. So the final summary of what we learned in this particular lecture is Cauchy stress, sigma is a second-order tensor. The element of stress tensor represents components of traction acting on three orthogonal planes according to a given Cartesian coordinate.   Sigma I j means j component of traction vector acting on the ith plane. Stress tensor sigma has three normal stress components and six shear stress components. But based on equilibrium,   there are six independent stresses three normal and three shear stresses.  All the stress components are acting at a point that is very relevant and which is very important. The components of sigma depend on the coordinate axis,   please note here as such sigma is not dependent on the coordinate axis,   but the components of sigma, I mean to say sigma x, sigma xy those are the components or the traction vector components, they are dependent on the coordinate axis. So there is a distinction that needs to be very clear, one should not get confused with sigma as a whole and the components of sigma. Sigma as a whole is not dependent on any axis,   but the components of sigma keep changing, but the overall sigma representation of internal force remains the same depending on the reference axis, and the component’s magnitude value keeps changing.   Stress tensor sigma at any point in the body defines the internal force distribution of a body.   So this is all about this particular lecture, which we will see in the next lecture.As found on YouTubeAnimationStudio ꆛ☣ꐕ Be The “Middle Man” And Profit With AnimationStudio Agency License. Here’s How You Can Earn $100, $200, or even $300 For Every Video You Create With AnimationStudio… Activate Your Profit Machine With The Agency License … $197/month For Just $67 One Time Payment