Hi Professor Barkey, I've actually used this concepts in Engineering appliacations so those are sort of descrete forms of applying the concepts not closed soutions. I've been told though, they have basis in theory of plasticity, I cannot completly relate it when instead of a yield surface composed of different stresses we're presented with a yield surface composed of axial load and bending moment in the other axis, isn't that stress in the same direction? at the end bending stress is nothing but an axial stress distribution across the depth of the member. In order to have a yield surface isn't necessary to have stresses in different axial directions let's say sigma x and sigma y in other axis? Any suggestion to understand this relationships taking them from stresses to forces for non linear structural anaysis problems? Thank you so much.
Hi, I do not have any direct experience with that. You might look at limit load analysis of structures. You can define a failure envelope based on loading instead of stress, but it would not be clear to me how to relate that to something like deformation as we can relate stress to strains via the normality flow rule.
@@mbarkey.mechanics so the faluire envelope instead of a stress envolpe it's possible? Well, the way I've seen it to be related via flow rule, it's to use a plastic deformation matrix which depends on the stiffness matix of the structure, it's basically gradient G times some landa factor which defines the plastic deformation, so I see some similarities. The total displacement/deformation vector depends of a tangent stiffness matrix which is the sum of a plastic stiffness matrix reduction (which is affected by the plastic deformation) plus the elastic one. It's an incremental load anlysis. If that makes sense to you I would comfortable with the idea of using it that way.
@@ing.erickosorio2887 Hi, I can't really say and I am not fully familiar with that process. The normality flow rule really depends on convexity of the yield surface, and I don't thing a failure envelope approach could guarantee convexity. But I do not have enough information about the approach you describe to have any opinion (and no time to research it these days). Take care!
Firstly, I would like to thank you for awesome lectures. I have a doubt. What is the order of unit normal? Is there any literature available where unit normal has been determined?
@@mbarkey.mechanics Definitely sir. A unit normal can be determined by gradient of the function. But the yield surface formed is a hyper surface (or a hyper sphere). Thus the dimension of unit normal would be 9*1. How do we handle such situation in an Euclidean space.
@@chandrabhansingh3977 Everything translates to any number of finite dimensions just as it does from 2D to 3D. The only thing lost is the ability to visualize it in the native N-dimensional space. If you follow these lectures you will see worked out examples. Look up the distance formula for 2D, 3D, and then N-dimensional space for another example.
thank you for a lecture, really helpful. I am new to plasticity theory, your series on plasticity have given me a good starting point. I would like to understand it in detail, do you recommend any book to begin with?
There are many areas of plasticity: yield surface (like in my notes); metal forming; dynamic plasticity, etc. Each area has some good books. I like the book by Jacob Lubliner (can be found for free on the internet) and also the classical book by Rodney Hill (can be found at low cost). If you read those two, then you can read pretty much anything else. To really be at the front of the knowledge in the area, you will want to read journal articles (e.g. International Journal of Plasticity) in your area of interest, and then even go to some conferences.
Thank you sir for this lecture series. I have one doubt. At first u mentioned that we are looking for a linear relationshio between epsilon p and sigma. But it turns out that d lamda is a function of vonmises stress, which again is a function of the current stress state. Could you please correct me if my understanding is wrong.?
Hi Professor Barkey, I've actually used this concepts in Engineering appliacations so those are sort of descrete forms of applying the concepts not closed soutions. I've been told though, they have basis in theory of plasticity, I cannot completly relate it when instead of a yield surface composed of different stresses we're presented with a yield surface composed of axial load and bending moment in the other axis, isn't that stress in the same direction? at the end bending stress is nothing but an axial stress distribution across the depth of the member. In order to have a yield surface isn't necessary to have stresses in different axial directions let's say sigma x and sigma y in other axis? Any suggestion to understand this relationships taking them from stresses to forces for non linear structural anaysis problems?
Thank you so much.
Hi, I do not have any direct experience with that. You might look at limit load analysis of structures. You can define a failure envelope based on loading instead of stress, but it would not be clear to me how to relate that to something like deformation as we can relate stress to strains via the normality flow rule.
@@mbarkey.mechanics so the faluire envelope instead of a stress envolpe it's possible? Well, the way I've seen it to be related via flow rule, it's to use a plastic deformation matrix which depends on the stiffness matix of the structure, it's basically gradient G times some landa factor which defines the plastic deformation, so I see some similarities. The total displacement/deformation vector depends of a tangent stiffness matrix which is the sum of a plastic stiffness matrix reduction (which is affected by the plastic deformation) plus the elastic one. It's an incremental load anlysis. If that makes sense to you I would comfortable with the idea of using it that way.
@@ing.erickosorio2887 Hi, I can't really say and I am not fully familiar with that process. The normality flow rule really depends on convexity of the yield surface, and I don't thing a failure envelope approach could guarantee convexity. But I do not have enough information about the approach you describe to have any opinion (and no time to research it these days). Take care!
Firstly, I would like to thank you for awesome lectures. I have a doubt. What is the order of unit normal? Is there any literature available where unit normal has been determined?
I don't think I understand the question. You can find a unit normal vector for any smooth, continuous, convex surface.
@@mbarkey.mechanics Definitely sir. A unit normal can be determined by gradient of the function. But the yield surface formed is a hyper surface (or a hyper sphere). Thus the dimension of unit normal would be 9*1. How do we handle such situation in an Euclidean space.
@@chandrabhansingh3977 Everything translates to any number of finite dimensions just as it does from 2D to 3D. The only thing lost is the ability to visualize it in the native N-dimensional space. If you follow these lectures you will see worked out examples. Look up the distance formula for 2D, 3D, and then N-dimensional space for another example.
thank you for a lecture, really helpful. I am new to plasticity theory, your series on plasticity have given me a good starting point. I would like to understand it in detail, do you recommend any book to begin with?
There are many areas of plasticity: yield surface (like in my notes); metal forming; dynamic plasticity, etc. Each area has some good books. I like the book by Jacob Lubliner (can be found for free on the internet) and also the classical book by Rodney Hill (can be found at low cost). If you read those two, then you can read pretty much anything else. To really be at the front of the knowledge in the area, you will want to read journal articles (e.g. International Journal of Plasticity) in your area of interest, and then even go to some conferences.
@@mbarkey.mechanics thank you sir 😃. ....
Thank you sir for this lecture series.
I have one doubt.
At first u mentioned that we are looking for a linear relationshio between epsilon p and sigma. But it turns out that d lamda is a function of vonmises stress, which again is a function of the current stress state.
Could you please correct me if my understanding is wrong.?