a direct quasi-linear utility function, using duality theory. and test whether the statistical demand function is integrable by exploring the. In turn, a utility function tells us the utility associated with each good x ∈ X, and is .. Quasilinear preferences are linear in x2, so the marginal utility is constant. is quasilinear in commodity 1 if there is a utility function that represents it in the .. In general, there is a calculus test to determine if a utility function satisfies the .
BME Determining Constants for the Quasilinear Viscoelastic Model
Indifference curves with the property that the MRS depends only on free time. Figure 1 Indifference curves with the property that the MRS depends only on free time.
A utility function with the property that the marginal rate of substitution MRS between and depends only on is: This is called a quasi-linear function because utility is linear in and some function of. We now show that this utility function has the required property.
It may be found by the formula we derived in the earlier Leibniz: In this case, andso The same result can be obtained directly, without using the general formula. Each indifference curve is of the form orwhere is a constant. Therefore along an indifference curve.
The curve slopes downwards and the absolute value of the slope is. Thus the MRS is a function of alone, as we wished to prove. In Figure 1, the indifference curves have the usual property of diminishing MRS, flattening as you move to the right. For this to happen, must fall as increases. Because indifference curves are of the formany two of them differ by a constant vertical distance, as you can see in Figure 1. The reason why the curves in the diagram bunch together horizontally at large values of is simply that they are steeper there.
Using a utility function of this form means that we are making a restrictive assumption about preferences, but it has a very useful implication. It has the capability of modeling materials with time dependent viscoelastic behavior that undergo large deformation.
Quasilinear utility - Wikipedia
As with any constitutive equation, we must be able to fit the constants in the model. To fully fit this constitutive model, we need to perform time dependent stress relaxation tests and fit the constants of the function that characterizes stress relaxation as well as large deformation. Difficulties in fitting the quasilinear model for stress relaxation tests were previously complicated by the need to have a very fast ramp to the applied strain.
Recently, Abramowitch and Woo published a method to fit the QLV model using a ramp time with finite speed.
Abramowitch, SD and Woo, SLY "An improved method to analyze the stress relaxation of ligaments following a finite ramp time based on the quasi-linear viscoelastic theory", Journal of Biomechanical Engineering, Therefore, we adopt a simpler reduced relaxation function proposed by Toms et al. At the end of the section, you should: Be better able to understand QLV theory 2. Understand the experimental tests necessary to determine QLV constants 3.Linear Viscoelastic Materials & Models
QLV Theory Revisited As noted previously, QLV theory models the viscoelastic response of a material based on a stress relaxation function and the instantaneous stress resulting from a ramp strain as: G t is defined as: The complete stress history at any time t is then the convolution integral: We can assume that t starts at 0 instead of negative infinity for the experimental situaiton.
The reduced relaxation function proposed by Toms et al is: The instantaneous stress response is assumed to be represent through the nonlinear elastic relationship: