>is easier to read the independent values on the horizontal scale and the dependent ones on the vertical one.<<br>
The independent variable is the pressure you apply, it plots on X
The dependent variable is the deformation, it plots on Y
You should worry most about the data you have collected, many of them show an unexplainable pressure cycle. Some you can defend and I can accept, typically the one I have worked. What you are not saying is about the pressure cycle you apply or attempt to apply [I mean the ones that makes sense]. That pressure cycle pattern, does it comes from a standard in that kind of testing or does it come from some invention. Up to the fracture, if you apply a linear pressure, it will surely fit well.
The other point is that you will not interpolate via "linterp" or a bspline [interp] on the non ascending part of the pressure cycle, unless the data are split and transformed . The attached demonstrates how to interpolate the ascending viscoelastic sites. The part after [polyline] is for a total pressure cycle in parametric form.
>My main goal is to see the best curve to fit experimental data. That is why, in the first place, I'm studying what others have proposed (i.e. Wang et. al). <<br>
The best fit up to fracture will largely depend upon the ascending pressure . When you plot sin(x), x is supposed linear. Sometimes transforming the independent variable helps the fit. You refer to Wang et al. but you haven't attached the PDF or else link to what you are reading, just to make sure it is understandable, and technically correct. Does Wang give the form of the pressure cycle or is it an invention. Why don't you make it linear up to the fracture ?
The last plot is a 2/2 rational approximation.
jmG
The independent variable is the pressure you apply, it plots on X
The dependent variable is the deformation, it plots on Y
You should worry most about the data you have collected, many of them show an unexplainable pressure cycle. Some you can defend and I can accept, typically the one I have worked. What you are not saying is about the pressure cycle you apply or attempt to apply [I mean the ones that makes sense]. That pressure cycle pattern, does it comes from a standard in that kind of testing or does it come from some invention. Up to the fracture, if you apply a linear pressure, it will surely fit well.
The other point is that you will not interpolate via "linterp" or a bspline [interp] on the non ascending part of the pressure cycle, unless the data are split and transformed . The attached demonstrates how to interpolate the ascending viscoelastic sites. The part after [polyline] is for a total pressure cycle in parametric form.
>My main goal is to see the best curve to fit experimental data. That is why, in the first place, I'm studying what others have proposed (i.e. Wang et. al). <<br>
The best fit up to fracture will largely depend upon the ascending pressure . When you plot sin(x), x is supposed linear. Sometimes transforming the independent variable helps the fit. You refer to Wang et al. but you haven't attached the PDF or else link to what you are reading, just to make sure it is understandable, and technically correct. Does Wang give the form of the pressure cycle or is it an invention. Why don't you make it linear up to the fracture ?
The last plot is a 2/2 rational approximation.
jmG
On 5/4/2010 3:21:17 PM, rijackson wrote:
>On 5/4/2010 2:14:47 PM, dvmoldovan
>wrote:
>>Yes, richard that is what I
>>need. When talking about
>>column 2 you are referring
>>to..(what column/ in which
>>sheet?).
>
>It is getting hard to keep track of
>everything! Set c=2 in this sheet to see
>the data I am referring to:
>
>http://collab.mathsoft.com/~Mathcad2000/
>read?134452,11#134462
>
>Richard
________________________________
The Minerr Paul W. was passed in two styles: the short style as in the work sheet in reference, and the NULL style. The fastest when it works is the short style. However many fits will not be possible in short style, so here is the NULL style. The vector of initial values must be collected in vector, this is so because lots of fit can't go direct LM but a good approximation is obtained sometimes CG, sometimes QN ... thus much easier to plug coeff's. And because there is always an "AND", too many fits can only be done manual, thus you must be able to collect quite a few fits in vector.
In Mathcad it's best practice to respect the position of the arguments and parameters. Nothing wrong y(x,A,B,C,D) but it's not correct and creates lots of problems especially in collaboration ... it should be y(A,B,C,D,x).
Same remark about ORIGIN that should be left 0.
To fit a "normalised rational", only the first term need be valued, generally the value 1 is OK. Not detrimental in this 0...1 range of 'x' but it will soon fail with a much higher range of 'x' because if the higher order monomials are valued it pushes the solver out of capability.
Nothing wrong with the work sheet in reference, but not a model to follow, at least not for beginners learning "curve fitting" in Mathcad.
jmG
>On 5/4/2010 2:14:47 PM, dvmoldovan
>wrote:
>>Yes, richard that is what I
>>need. When talking about
>>column 2 you are referring
>>to..(what column/ in which
>>sheet?).
>
>It is getting hard to keep track of
>everything! Set c=2 in this sheet to see
>the data I am referring to:
>
>http://collab.mathsoft.com/~Mathcad2000/
>read?134452,11#134462
>
>Richard
________________________________
The Minerr Paul W. was passed in two styles: the short style as in the work sheet in reference, and the NULL style. The fastest when it works is the short style. However many fits will not be possible in short style, so here is the NULL style. The vector of initial values must be collected in vector, this is so because lots of fit can't go direct LM but a good approximation is obtained sometimes CG, sometimes QN ... thus much easier to plug coeff's. And because there is always an "AND", too many fits can only be done manual, thus you must be able to collect quite a few fits in vector.
In Mathcad it's best practice to respect the position of the arguments and parameters. Nothing wrong y(x,A,B,C,D) but it's not correct and creates lots of problems especially in collaboration ... it should be y(A,B,C,D,x).
Same remark about ORIGIN that should be left 0.
To fit a "normalised rational", only the first term need be valued, generally the value 1 is OK. Not detrimental in this 0...1 range of 'x' but it will soon fail with a much higher range of 'x' because if the higher order monomials are valued it pushes the solver out of capability.
Nothing wrong with the work sheet in reference, but not a model to follow, at least not for beginners learning "curve fitting" in Mathcad.
jmG
... also, the principle of solving is to equate to 0 [zero]. This goes behind the scene of the symbolic solve and the root and ... Thus respecting the principle is the best procedure. On the other hand, equating to zero is sometimes an heresy that's much easier to manipulate on the right side.
jmG
jmG
On 5/5/2010 6:13:05 PM, bones7xx wrote:
>I don't see why the model
>selected is so complicated.
>The data is noisy and any
>least squares results are
>subject to speculation. At
>the end of this document is a
>simple Thiele run.
______________________________
Can't be more right about "complicated model". It was selected to demonstrate additional constrains and refined fit and the inactive Minerr on one of the coefficients... 3 birds one stone. This is the final fit.
Sorry for Wang model !
jmG
>I don't see why the model
>selected is so complicated.
>The data is noisy and any
>least squares results are
>subject to speculation. At
>the end of this document is a
>simple Thiele run.
______________________________
Can't be more right about "complicated model". It was selected to demonstrate additional constrains and refined fit and the inactive Minerr on one of the coefficients... 3 birds one stone. This is the final fit.
Sorry for Wang model !
jmG
On 5/9/2010 2:38:14 PM, dvmoldovan wrote:
>First, thank you all for your
>time and energy put to my
>rescue. Although your answers
>may sometimes have been not
>what I hoped for (initially),
>you all pointed out several
>flaws in my sheet (and data).
>With you permission I'll start
>over (still based on what has
>been posted here so far).
>Second, let's assume the
>following :
>1. from experiments one has
>gathered data for strains
>(column 1 - microns) and the
>response in stress from the
>material (column 0 - newtons)
>2. there is one function to
>link them together (as
>proposed by Wang)
>
>The question is, how can I
>achieve the best fit between
>experiments and Wang?
>
>I attached my latest work, but
>it doesn't feel right due to
>that inflection point...
>
>Please disregard erratic data,
>discontinuities and others.
>I'm trying to understand the
>principle and the formulas it
>should use before collecting
>any more data...
>
>Kind regards, D.
____________________________
"1. from experiments one has gathered data for strains (column 1 - microns) and the response in stress from the material (column 0 - newtons)"
==> Total non sense: strain is a calculated ratio, dimensionless.
I don't understand negative pressure unless the sensors have inverse polarity or the calibration is zeroed at 2500. But you can see that the project is reversed. So, we are back to square 0, i.e: read the pressure from a dial meter and the deformation from a dial meter too, with much less measurements.
During testing of a material sample, the stress�strain curve is a graphical representation of the relationship between stress, derived from measuring the load applied on the sample, and strain, derived from measuring the deformation of the sample, i.e. elongation, compression, or distortion.
In continuum mechanics, stress is a measure of the average force per unit area of a surface within a deformable body on which internal forces act. It is a measure of the intensity of the internal forces acting between particles of a deformable body across imaginary internal surfaces [2]. These internal forces are produced between the particles in the body as a reaction to external forces applied on the body. External forces are either surface forces or body forces. Because the loaded deformable body is assumed as a continuum, these internal forces are distributed continuously within the volume of the material body, i.e., the stress distribution in the body is expressed as a piecewise continuous function of space coordinates and time.
The SI unit for stress is the pascal (symbol Pa), which is equivalent to one newton (force) per square meter (unit area). The unit for stress is the same as that of pressure, which is also a measure of force per unit area.
Strain is the geometrical measure of deformation representing the relative displacement between particles in the material body. It measures how much a given displacement differs locally from a rigid-body displacement.[1] Strain defines the amount of stretch or compression along a material line elements or fibers, the normal strain, and the amount of distortion associated with the sliding of plane layers over each other, the shear strain, within a deforming body.[2] Strain is a dimensionless quantity, which can be expressed as a decimal fraction, a percentage or in parts-per notation. This could be applied by elongation, shortening, or volume changes, or angular distortion.
The Cauchy strain or engineering strain is expressed as the ratio of total deformation to the initial dimension of the material body in which the forces are being applied. The engineering normal strain or engineering extensional strain e of a material line element or fiber axially loaded is expressed as the change in length �L per unit of the original length L of the line element or fibers. The normal strain is positive if the material fibers are stretched or negative if they are compressed. Thus, we have
where is the engineering normal strain, L is the original length of the fiber and is the final length of the fiber.
The engineering shear strain is defined as the change in the angle between two material line elements initially perpendicular to each other in the undeformed or initial configuration
jmG
>First, thank you all for your
>time and energy put to my
>rescue. Although your answers
>may sometimes have been not
>what I hoped for (initially),
>you all pointed out several
>flaws in my sheet (and data).
>With you permission I'll start
>over (still based on what has
>been posted here so far).
>Second, let's assume the
>following :
>1. from experiments one has
>gathered data for strains
>(column 1 - microns) and the
>response in stress from the
>material (column 0 - newtons)
>2. there is one function to
>link them together (as
>proposed by Wang)
>
>The question is, how can I
>achieve the best fit between
>experiments and Wang?
>
>I attached my latest work, but
>it doesn't feel right due to
>that inflection point...
>
>Please disregard erratic data,
>discontinuities and others.
>I'm trying to understand the
>principle and the formulas it
>should use before collecting
>any more data...
>
>Kind regards, D.
____________________________
"1. from experiments one has gathered data for strains (column 1 - microns) and the response in stress from the material (column 0 - newtons)"
==> Total non sense: strain is a calculated ratio, dimensionless.
I don't understand negative pressure unless the sensors have inverse polarity or the calibration is zeroed at 2500. But you can see that the project is reversed. So, we are back to square 0, i.e: read the pressure from a dial meter and the deformation from a dial meter too, with much less measurements.
During testing of a material sample, the stress�strain curve is a graphical representation of the relationship between stress, derived from measuring the load applied on the sample, and strain, derived from measuring the deformation of the sample, i.e. elongation, compression, or distortion.
In continuum mechanics, stress is a measure of the average force per unit area of a surface within a deformable body on which internal forces act. It is a measure of the intensity of the internal forces acting between particles of a deformable body across imaginary internal surfaces [2]. These internal forces are produced between the particles in the body as a reaction to external forces applied on the body. External forces are either surface forces or body forces. Because the loaded deformable body is assumed as a continuum, these internal forces are distributed continuously within the volume of the material body, i.e., the stress distribution in the body is expressed as a piecewise continuous function of space coordinates and time.
The SI unit for stress is the pascal (symbol Pa), which is equivalent to one newton (force) per square meter (unit area). The unit for stress is the same as that of pressure, which is also a measure of force per unit area.
Strain is the geometrical measure of deformation representing the relative displacement between particles in the material body. It measures how much a given displacement differs locally from a rigid-body displacement.[1] Strain defines the amount of stretch or compression along a material line elements or fibers, the normal strain, and the amount of distortion associated with the sliding of plane layers over each other, the shear strain, within a deforming body.[2] Strain is a dimensionless quantity, which can be expressed as a decimal fraction, a percentage or in parts-per notation. This could be applied by elongation, shortening, or volume changes, or angular distortion.
The Cauchy strain or engineering strain is expressed as the ratio of total deformation to the initial dimension of the material body in which the forces are being applied. The engineering normal strain or engineering extensional strain e of a material line element or fiber axially loaded is expressed as the change in length �L per unit of the original length L of the line element or fibers. The normal strain is positive if the material fibers are stretched or negative if they are compressed. Thus, we have
where is the engineering normal strain, L is the original length of the fiber and is the final length of the fiber.
The engineering shear strain is defined as the change in the angle between two material line elements initially perpendicular to each other in the undeformed or initial configuration
jmG
On 5/9/2010 6:32:06 PM, jmG wrote:
>... more graphs
>
>jmG
________________
... and more functions to spend a good evening
jmG
>... more graphs
>
>jmG
________________
... and more functions to spend a good evening
jmG
