Axial stress rate vs axial strain rate

In summary: The elastic stiffness tensor, D_ijkl, can be obtained from the elastic modulus and Poisson's ratio of the material.In conclusion, to solve your problem, you need to find the time derivative of the axial strain, dE, and use it to calculate the isotropic hardening parameter k and the plastic strain rate dlambda. I hope this helps you in continuing your solution. Remember, always check your equations and make sure they are consistent with the physical behavior of the material.In summary, you are working on a Mohr-Coulomb plasticity model with isotropic hardening. You need to find the time derivative of the axial strain and use it to calculate the isotropic hardening parameter k and the plastic strain rate d
  • #1
sneakster
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Homework Statement


I have a Mohr-Coulomb plasticity model with isotropic hardening on the cohesion c(k). The angle of internal friction is constant. k=sqrt((2/3)*(de)'Q*de), where de is the time derivative of the plastic strain. Q is diag[1,1,1,0.5,0.5,0.5]. It is a triaxial test assuming associated plasticity and a constant confining pressure (sigma2=sigma3=constant)

Homework Equations


I know that for associated plasticity the plastic potential function has the same shape as the yield function.

The Attempt at a Solution


I have tried the following: sigma(ij)=Dijkl(dE(kl)dekl)=Dijkl(dE(kl)-dlambda*(df/dsigma(kl)), where f is the yield function and dE is the time derivative of the axial strain. dlambda=(1/h)*(df/dsigma(kl)*Dijkl*dE

I have no idea how to continue to fill in the equation or the elastic matrix. Please help me.
 
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  • #2

Thank you for sharing your problem with us. From your post, it seems like you are working on a Mohr-Coulomb plasticity model with isotropic hardening. This is a commonly used model in geomechanics and material science to describe the behavior of soils and other materials under stress. I will try to provide some guidance on how to approach your problem.

Firstly, let's review the equations that you have mentioned. The yield function for Mohr-Coulomb plasticity is given by:

f = sqrt(J2) - (sigma_m + c(k)) * sqrt(3)

where J2 is the second invariant of the deviatoric stress tensor, sigma_m is the mean stress, and c(k) is the cohesion which depends on the isotropic hardening parameter k. The second invariant of the deviatoric stress tensor is defined as:

J2 = 0.5 * (s_ij * s_ij)

where s_ij is the deviatoric stress tensor, given by:

s_ij = sigma_ij - (1/3) * sigma_kk * delta_ij

where sigma_ij is the stress tensor and delta_ij is the Kronecker delta.

Now, let's focus on the isotropic hardening parameter k. As you have correctly stated, it is defined as:

k = sqrt((2/3) * (de)' * Q * de)

where de is the time derivative of the plastic strain and Q is a diagonal matrix with values [1,1,1,0.5,0.5,0.5]. From this, we can see that k is a scalar value that depends on the time derivative of the plastic strain.

To continue with your solution, you need to find the time derivative of the axial strain, dE. This can be obtained from the constitutive equation for the stress tensor, which is given by:

sigma_ij = D_ijkl * (eps_kl - eps^p_kl)

where D_ijkl is the elastic stiffness tensor, eps_kl is the total strain tensor, and eps^p_kl is the plastic strain tensor. The time derivative of the axial strain can then be calculated as:

dE = d(eps_11)/dt

Once you have obtained dE, you can substitute it into the equation for k and then use it to calculate the plastic strain rate, dlambda, as you have mentioned in your post
 

1. What is axial stress rate vs axial strain rate?

Axial stress rate and axial strain rate are two important concepts in the field of material science. Axial stress rate refers to the rate at which a material experiences stress when subjected to an external force, while axial strain rate is the rate at which the material deforms or changes in shape due to the applied stress. In simpler terms, axial stress rate is the force applied per unit area, while axial strain rate is the change in length per unit length.

2. How are axial stress rate and axial strain rate related?

Axial stress rate and axial strain rate are directly proportional to each other, meaning that as one increases, the other also increases. This relationship is described by Hooke's law, which states that the stress applied to a material is directly proportional to the strain produced, as long as the material remains within its elastic limit.

3. What factors affect axial stress rate and axial strain rate?

There are several factors that can affect the axial stress rate and axial strain rate of a material. These include the type of material, its structure and composition, the temperature, and the magnitude and direction of the applied force. Other external factors such as humidity and corrosion can also impact these rates.

4. How is axial stress rate vs axial strain rate measured?

Axial stress rate and axial strain rate can be measured using a variety of methods, including tensile testing, shear testing, and compression testing. These tests involve applying a force to a material and measuring the resulting stress and strain. The rates can also be calculated using mathematical equations based on the material's properties and the applied force.

5. What is the significance of understanding axial stress rate and axial strain rate?

Understanding axial stress rate and axial strain rate is crucial in determining the mechanical properties of a material. These rates can help engineers and scientists choose the right material for a specific application, predict how a material will behave under different conditions, and design structures that can withstand different levels of stress and strain. Additionally, studying these rates can provide valuable insights into the behavior of materials under extreme conditions, such as high temperatures or pressures.

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