A question has arised when I read a book about fracture mechanics recently.
This question is about the calculation of potential energy in "fixed-grips" (prescribed displacement) and "dead-load" (prescribed displacement).
Opinions in the book: In the condition of prescribed displacement, the potential energy is equal to the area between the displacement axis and the curve of load .vs. displacement, and has a positive value. In the condition of prescribed loading, the potential energy is equal to the area between the load axis and the curve of load .vs. displacement, and has a negative value.
Anyone can explain the above conclusion for me please?
Thank you!
potential energy of an elastic body subject to a constant force
Here is a way to think about the potential energy.
A constant force P can be represented by a dead weight. When the weight drops by a displacement u, the potential energy of the weight reduces by Pu.
When the weight is hung on an elastic body, the weight and the elastic body together form a system. The potential energy of the system is sum of the potential energy of the weight, -Pu, and the elastic energy of the body, U, namely,
Potential energy of the system = U - Pu.
It is this potential energy of the system that needs be minimized.
On the (P,u) plane, U is the area under the P-u curve, and Pu is the area of the rectangle.
In reply to potential energy of an elastic body subject to a constant force by Zhigang Suo
Further understanding along Prof. Zhigang's way
Hi, Prof. Suo,
Thanks for your explicit explanation.
Along your way, I've got the answer to my origin questions as follows:
As for the dead-load case, "U is the area under the P-u curve, and Pu is the area of the rectangle." Therefore, the potential energy = U - Pu <0, with its absolute value being the area between the P-u curve and P-axis on the P-u plane.
As for the prescribed-displacement case, the displacement along load P is equal to zero because the displacement has been fixed. In that case, the potential energy = U - Pu = U - P * 0 = U >0, equal to the area between P-u curve and u-axis.
Please help me to confirm whether my above understanding is right or not.
Thank you in advance!
In reply to Further understanding along Prof. Zhigang's way by Liu Jinxing
fixed force vs. fixed displacement
You are correct. To recap, when an elastic body is loaded by a weight (i.e., a fixed force P), the system consists of the body and the weight, and the energy to minimize is U - Pu.
When the elastic body is pulled to a fixed displacement, the system consists of the body alone, and the energy to minimize is U.
Potentials in general
As an added comment to Zhigang's remark, you should always recall what potentials are supposed to do. The negative gradient of a potential (with respect to motion) is supposed to give the force associated with a force-system. In the case Zhigang mentions, there is the dead-weight ( Potential = -height*weight ; -grad(Potential) wrt height = P) and the elastic system (Potential = 1/2 k * motion^2 ; -grad(Potential) wrt motion = -k * motion ). Thus the stationarity of the potential energy is nothing other than a statement of balance of forces in a system (for equilibrium).
Prof. Dr. Sanjay Govindjee
University of California, Berkeley
In reply to Potentials in general by Sanjay Govindjee
Another question
Thanks, Prof. Sanjay,
In the case of prescribed displacement, I still have difficulties in understanding the relationship between potential energy and force balance. I mean, in this case, the displacement is fixed instead of dead-weight. Can you please explain it to me?
In reply to Another question by Liu Jinxing
complementary energy method
Dear Jinxing
I think it's better to understand this from the view of complementary energy method. I'll try my best to address it clearly.Take the one dimensional problem for example, the complementary energy of the structure can be expressed:
Pi_c = 1/2C*P^2 - P*u
where, C is the compliance of the structure, P is the support reaction due to the prescribed displacement u. It's noted that P is the real variable in this situation. Minimize the total complementary energy, it leads to the following equation:
C*P = u
For the linear problem, compliance is the inverse of the stiffness of the structure, which reads C = 1/KSo, the above equation can be rewritten as:
Ku = P
These two equations can be seen as different expressions of the force balance.
As for the value of the potential energy under the displacements control case, since no applied is prescribed on structure, i.e. P = 0, the work of the external force F is zero.
To get a better understanding, I recommend you read some books about the variational methods in mechanics, such as, variational principle of elastic mechanics and its application(in Chinese 弹性力学的变分原理及其应用, 胡海昌), variational principle and finite element (in Chinese 变分原理与有限元，钱伟长), and the mechanics of materials written by Timoshenko also has a chapter that describe the energy method, which is very useful from my experience.
I hope this may help you understand this problem.
Teng zhang
some confusion on finite compliance
I am reading the fracture book written by T.L Anderson, I have some confusion on the derivation of the energy release rate for the structure with finite compliance.
The book (see page 115 for reference) says the fixed remote displacement deltaT prescribed in the structure can be given as:
deltaT = delta + C_M*P
where delta is the local load line displacement, P is the applied load and C_M is the compliance of the spring.
It can be seen that when C_M = 0, current load form corresponds to the pure displacement control, however, I couldn't understand well why the infinitely soft spring implies load control. If the C_M = ∞, how can we obtain the value of the applied load P from the above equation?
I hope someone can help understand this issue.
Thank you!
Teng zhang
I made a mistake
Dear Jinxing, Prof. Suo and Sanjay
I'm sorry that I have a misunderstanding of the complementary energy method (Principle of minimum complementary energy).
The Principle of minimum complementary energy is actually another statement of the of the compatibility condition and displacements boundary condition. The lecture note of linear Elasticity in Brown university presents a good description of this principle. http://www.engin.brown.edu/courses/en224/compenergy/compenergy.html.
In my previous comments, I take a wrong explaination of the equation C*P = u. I think this should be seen as the displacement BC with support reaction as variable. I try to express my thought as follows. The complementary energy is written as
Pi_c = integrator(B(stress)) - P*u.
where B is the denstiy of the complementary energy, P is the support reaction. In this simple example, the statically admissible stress is
stress = constant = P/A,
where A is the cross area. Thus the complementary energy can be rewritten as
Pi_c = 1/2C*P^2 - P*u.
For this problem, the compatibility condition satisfies naturaly, thus, the equation should be regarded as the displacement boundary condition.
Stress is dual variable of strain, on the other hand, stress functions can be seen as the dual variables of the displacements. If we use stress functions as the fundamental variables of the complementary energy, the Principle of minimum complementary energy leads to the typically stress function methods in solving elasticity problem, while, the Principle of minimum potential energy corresponds to the displacements methods in solving elasticity problem.
I'm not sure whether my understanding right or not, so, I hope to hear comments form Prof. Suo, Sanjay and others. Besides, Prof. Suo gives a simple but impressive explaination of the potential energy, I wonder if there is a similar way of thinking complementary energy?
Thanks in advance
Teng Zhang