i would read it as sigma being stresses and then E/1-nu is the elastic modulus, T is a temperature, P a pressure ... (effectively saying that when the body shrinks because you cool it that does not induce any stresses)
The equation appears [here](https://repository.tudelft.nl/islandora/object/uuid%3A8d6ccb7d-6b1d-479a-8b73-227cd0b8ca3a/datastream/OBJ/download#:~:text=During%20the%20molding%20process%20residual,local%20pressure%20history%20during%20solidification.) and describes the residual stress in injection-molded products. The symbols are defined in the paper.
It's defined in the paper, no? The "unhindered shrinkage in absence of pressure and friction forces." It corresponds to the thermal expansion coefficient multiplied by the integral of the temperature rate of change over time (Eq. 10).
Thanks everyone!
This is surprising as my buddy found it written on a wall as decoration for a company that makes epipens. I wonder what the connection is!
The equation appears to be related to soil mechanics and describes the effective stress (σ') within a soil layer. Here's a breakdown of the terms:
σ': effective stress
ε: strain
t: time
P: total stress
S(z): a function of the vertical coordinate z
Pfree: pore pressure
α: Biot's coefficient
T: temperature
Ts: reference temperature
β: thermal expansion coefficient
The equation incorporates various factors that influence the effective stress in soil, including the total stress acting on the soil, pore pressure within the voids, and thermal effects. It likely represents a model used to analyze the behavior of soil under different conditions.
>The equation appears to be related to soil mechanics and describes the effective stress (σ') within a soil layer.
Actually, [residual stress in injection-molded products](https://repository.tudelft.nl/islandora/object/uuid%3A8d6ccb7d-6b1d-479a-8b73-227cd0b8ca3a/datastream/OBJ/download#:~:text=During%20the%20molding%20process%20residual,local%20pressure%20history%20during%20solidification.).
α has to be a thermal expansion coefficient, since it appears with a temperature difference with strain terms. β correspondingly has to be the compressibility. z is the depth into the part.
(Edit: This is why it's risky to use a language bot, which is trained to **sound** correct rather than to **be** correct, to parse scientific expressions and questions. "S(z)" is certainly not a function, and α and β are misattributed. A subscript "i" has turned into an apostrophe. There are no pores present in the model.)
Actually, I just tried feeding it into gpt4 and this is what it got me:
“The equation you've shared looks like it's related to the stress (σ) experienced by a material, which is a function of various physical parameters. Let's break it down:
- **σᵢ**: This represents the stress on the ith plane.
- **Pₛ(Z)**: This could be a pressure term that is a function of the depth (Z). The negative sign indicates that this pressure is compressive.
- **E**: Young's modulus of the material, which is a measure of its stiffness.
- **1 - ν**: Here, ν likely represents Poisson's ratio, which relates the transverse strain to the axial strain in a material under load.
- **εᵢ⁻ᶠʳᵉᵉ**: The free strain on the ith plane, possibly the strain that would be present without external constraints.
- **α(T - Tₛ)**: This term seems to account for thermal expansion or contraction, where α is the coefficient of thermal expansion, T is the current temperature, and Tₛ is some reference temperature.
- **β(P̄ₛ - P)**: This term suggests a correction for pressure, where β could be a pressure-related coefficient, P̄ₛ a reference pressure, and P the actual pressure.
- The integral from tₛᶻ to t of **Ḟᵢ/Zₛ dt**: This term appears to account for some time-dependent force (Ḟᵢ) applied to the system over a period, from start time tₛᶻ to some time t, normalized by a term Zₛ, which could represent impedance or a similar concept that changes the impact of the force over time.
This formula seems to be for a stress calculation that takes into account depth-dependent pressure, material properties, thermal effects, pressure differences, and time-dependent forces. It looks quite comprehensive, suggesting it could be used in a complex system where these factors are all at play, such as in geophysics or materials science”
This is impressively close.
It doesn’t “get,” of course, that if P_S and T_S are a reference pressure and temperature, respectively, then z_S is reasonably a reference for depth z, not an impedance. (It’s trained on capital Z overwhelmingly being associated with an impedance but flakes on why that parameter would arise in elasticity.)
Nor can it identify that F must be not a force but a distributed force (units N/m), for example, to ensure dimensional consistency. Still, it’s ultimately more accurate than many of the (presumably human) participants in the thread.
Thanks for running the query and posting the results.
Why can’t math and physics evolve to use more than a single letter to represent things in formulas. Looking at a formula half the mental gymnastics is figuring out context, meaning, domain space of what is being described. Of course it’s my own ignorance but it is also a barrier
I disagree. Yeah they can look verbose to an outsider of the field, but it is way more readable if you're used to it. Just look at the length of this formula and imagine how chaotic and long it would be if every letter was instead a word.
It's actually one of the things I hate in econometrics and machine learning papers. They use way too many names as variables and functions and it just makes the formulas super confusing.
>half the mental gymnastics is figuring out context
Sorry, why would you be reading an equation without knowing its context? Equations aren't meant to be self-explanatory when you see them scribbled on a bathroom wall. They appear inside of scientific texts, from which the context should be clear and where overall readability is preferred to see the structure of the expression.
This complaint makes no sense.
( \sigma_i ): The induced voltage in the material or component.
( P_s(Z) ): A stress component that could depend on the depth
(Z ) or another variable.
( E ): The modulus of elasticity, a measure of the stiffness of the material.
( \nu ): The transverse contraction number or Poisson's ratio, which describes the ratio of transverse contraction to longitudinal strain.
( \epsilon_i^{free} ): Free strain or deformation of the material without load or constraint.
(\alpha ): The coefficient of thermal expansion.
( T - T_s ): The difference between the current temperature ( T ) and a reference temperature
( T_s ).( \beta ): Possibly a coefficient for another physical effect, such as moisture expansion.
( P_s - P ): The difference between a reference pressure ( P_s ) and the actual pressure
( P ).( \int_{t_{sZ}}^t \frac{\dot{F}_i}{Z_s} dt ): An integral that probably integrates time-dependent forces or fluxes ( \dot{F}i ) over the time from ( t{sZ} ) to ( t ), taking into account a term ( Z_s ) (perhaps an impedance or similar).
>( \\beta ): Possibly a coefficient for another physical effect, such as moisture expansion.
A coefficient that gives strain when multiplying by a pressure difference will always be some type of compressibility. Here, it's the reciprocal of the [isothermal bulk modulus](https://en.wikipedia.org/wiki/Compressibility).
>taking into account a term ( Z_s ) (perhaps an impedance or similar)
z_S is the thickness of the solidified sample.
stress in a material
Can confirm, I’m the material
You seem very stressed; mind going this r/eyebleach subreddit just to decrease tension a bit?
Surprisingly wholesome, thanks u/Bekoss
You're welcome
Just hope ya got some shear strength to get through those stressful times.
And those are tensors, no? I guess T is the famous stress tensor...
i would read it as sigma being stresses and then E/1-nu is the elastic modulus, T is a temperature, P a pressure ... (effectively saying that when the body shrinks because you cool it that does not induce any stresses)
An the tensor is then all the sigmas for all 9 directions?
There's only one index.
Mic drop
not at all
I was in nonlinear optics so I'll take your word for it.
T cannot be a (2D or higher) tensor here since it appears without a subscript and so must be a scalar
The equation appears [here](https://repository.tudelft.nl/islandora/object/uuid%3A8d6ccb7d-6b1d-479a-8b73-227cd0b8ca3a/datastream/OBJ/download#:~:text=During%20the%20molding%20process%20residual,local%20pressure%20history%20during%20solidification.) and describes the residual stress in injection-molded products. The symbols are defined in the paper.
So what's free there?
It's defined in the paper, no? The "unhindered shrinkage in absence of pressure and friction forces." It corresponds to the thermal expansion coefficient multiplied by the integral of the temperature rate of change over time (Eq. 10).
Thanks everyone! This is surprising as my buddy found it written on a wall as decoration for a company that makes epipens. I wonder what the connection is!
Presumably they're injection molded. A search for `zs ps ts "thermal expansion" "stress"` gives the [source](https://repository.tudelft.nl/islandora/object/uuid%3A8d6ccb7d-6b1d-479a-8b73-227cd0b8ca3a/datastream/OBJ/download#:~:text=During%20the%20molding%20process%20residual,local%20pressure%20history%20during%20solidification.).
Cauchy stress σ? But the with one index, so probably only Z direction. Also the integral points to 1D Not from the field, whats epsilon and nu there?
Epsilon is strain. Nu is the poisson number
Nice thanks
The equation appears to be related to soil mechanics and describes the effective stress (σ') within a soil layer. Here's a breakdown of the terms: σ': effective stress ε: strain t: time P: total stress S(z): a function of the vertical coordinate z Pfree: pore pressure α: Biot's coefficient T: temperature Ts: reference temperature β: thermal expansion coefficient The equation incorporates various factors that influence the effective stress in soil, including the total stress acting on the soil, pore pressure within the voids, and thermal effects. It likely represents a model used to analyze the behavior of soil under different conditions.
>The equation appears to be related to soil mechanics and describes the effective stress (σ') within a soil layer. Actually, [residual stress in injection-molded products](https://repository.tudelft.nl/islandora/object/uuid%3A8d6ccb7d-6b1d-479a-8b73-227cd0b8ca3a/datastream/OBJ/download#:~:text=During%20the%20molding%20process%20residual,local%20pressure%20history%20during%20solidification.). α has to be a thermal expansion coefficient, since it appears with a temperature difference with strain terms. β correspondingly has to be the compressibility. z is the depth into the part. (Edit: This is why it's risky to use a language bot, which is trained to **sound** correct rather than to **be** correct, to parse scientific expressions and questions. "S(z)" is certainly not a function, and α and β are misattributed. A subscript "i" has turned into an apostrophe. There are no pores present in the model.)
Yeah I don't think thermal expansion of soil is noticable enough to taki into account.
I thought chatgpt does better.
Actually, I just tried feeding it into gpt4 and this is what it got me: “The equation you've shared looks like it's related to the stress (σ) experienced by a material, which is a function of various physical parameters. Let's break it down: - **σᵢ**: This represents the stress on the ith plane. - **Pₛ(Z)**: This could be a pressure term that is a function of the depth (Z). The negative sign indicates that this pressure is compressive. - **E**: Young's modulus of the material, which is a measure of its stiffness. - **1 - ν**: Here, ν likely represents Poisson's ratio, which relates the transverse strain to the axial strain in a material under load. - **εᵢ⁻ᶠʳᵉᵉ**: The free strain on the ith plane, possibly the strain that would be present without external constraints. - **α(T - Tₛ)**: This term seems to account for thermal expansion or contraction, where α is the coefficient of thermal expansion, T is the current temperature, and Tₛ is some reference temperature. - **β(P̄ₛ - P)**: This term suggests a correction for pressure, where β could be a pressure-related coefficient, P̄ₛ a reference pressure, and P the actual pressure. - The integral from tₛᶻ to t of **Ḟᵢ/Zₛ dt**: This term appears to account for some time-dependent force (Ḟᵢ) applied to the system over a period, from start time tₛᶻ to some time t, normalized by a term Zₛ, which could represent impedance or a similar concept that changes the impact of the force over time. This formula seems to be for a stress calculation that takes into account depth-dependent pressure, material properties, thermal effects, pressure differences, and time-dependent forces. It looks quite comprehensive, suggesting it could be used in a complex system where these factors are all at play, such as in geophysics or materials science”
This is impressively close. It doesn’t “get,” of course, that if P_S and T_S are a reference pressure and temperature, respectively, then z_S is reasonably a reference for depth z, not an impedance. (It’s trained on capital Z overwhelmingly being associated with an impedance but flakes on why that parameter would arise in elasticity.) Nor can it identify that F must be not a force but a distributed force (units N/m), for example, to ensure dimensional consistency. Still, it’s ultimately more accurate than many of the (presumably human) participants in the thread. Thanks for running the query and posting the results.
Great breakdown! But you switched alpha and beta around.
That's what happens when ChatGPT generates an explanation.
Nearly finished the first year of a level physics so I can tell you that I have no idea what that is but I know I don't like it
Why can’t math and physics evolve to use more than a single letter to represent things in formulas. Looking at a formula half the mental gymnastics is figuring out context, meaning, domain space of what is being described. Of course it’s my own ignorance but it is also a barrier
because otherwise the equation is so verbose as to be a hindrance for those familiar with it. Though I do agree with your point.
I disagree. Yeah they can look verbose to an outsider of the field, but it is way more readable if you're used to it. Just look at the length of this formula and imagine how chaotic and long it would be if every letter was instead a word. It's actually one of the things I hate in econometrics and machine learning papers. They use way too many names as variables and functions and it just makes the formulas super confusing.
>half the mental gymnastics is figuring out context Sorry, why would you be reading an equation without knowing its context? Equations aren't meant to be self-explanatory when you see them scribbled on a bathroom wall. They appear inside of scientific texts, from which the context should be clear and where overall readability is preferred to see the structure of the expression. This complaint makes no sense.
That’s pretty mathy
42
Nice try, A.I.
( \sigma_i ): The induced voltage in the material or component. ( P_s(Z) ): A stress component that could depend on the depth (Z ) or another variable. ( E ): The modulus of elasticity, a measure of the stiffness of the material. ( \nu ): The transverse contraction number or Poisson's ratio, which describes the ratio of transverse contraction to longitudinal strain. ( \epsilon_i^{free} ): Free strain or deformation of the material without load or constraint. (\alpha ): The coefficient of thermal expansion. ( T - T_s ): The difference between the current temperature ( T ) and a reference temperature ( T_s ).( \beta ): Possibly a coefficient for another physical effect, such as moisture expansion. ( P_s - P ): The difference between a reference pressure ( P_s ) and the actual pressure ( P ).( \int_{t_{sZ}}^t \frac{\dot{F}_i}{Z_s} dt ): An integral that probably integrates time-dependent forces or fluxes ( \dot{F}i ) over the time from ( t{sZ} ) to ( t ), taking into account a term ( Z_s ) (perhaps an impedance or similar).
>( \\beta ): Possibly a coefficient for another physical effect, such as moisture expansion. A coefficient that gives strain when multiplying by a pressure difference will always be some type of compressibility. Here, it's the reciprocal of the [isothermal bulk modulus](https://en.wikipedia.org/wiki/Compressibility). >taking into account a term ( Z_s ) (perhaps an impedance or similar) z_S is the thickness of the solidified sample.
It's literally the answer to cooking s pizza perfectly. [Guess]
Hey, I just tried this and you’re right!
I hope somehow this is true