**Derive the Viscoelastic Equation for the Four-Element Model**In viscoelasticity, the four-element model, also known as the Burger’s model, is used to describe the behavior of materials that exhibit both viscous and elastic characteristics when undergoing deformation.This model consists of two main components connected in series:1. **Kelvin-Voigt Model**: A spring and dashpot in parallel.2. **Maxwell Model**: A spring and dashpot in series.### Derivation Steps:1. **Model Components**: - *Maxwell Arm*: Includes a spring (\(E_1\)) and dashpot (\(\eta_1\)). - *Kelvin-Voigt Arm*: Includes a spring (\(E_2\)) and dashpot (\(\eta_2\)).2. **Constitutive Equations**: - For the Maxwell Model: \[ \sigma(t) = E_1 \cdot \varepsilon_1(t) + \eta_1 \cdot \frac{d\varepsilon_1}{dt} \] - For the Kelvin-Voigt Model: \[ \sigma(t) = E_2 \cdot \varepsilon_2(t') + \eta_2 \cdot \frac{d\varepsilon_2}{dt} \]3. **Connection in Series** means: \[ \varepsilon(t) = \varepsilon_1(t) + \varepsilon_2(t') \] where \(\varepsilon(t)\) is the total strain.4. **Overall Viscoelastic Equation**: - By combining the contributions from both components, the complete equation is derived to predict the viscoelastic response of the four-element model.The final form describes the stress-strain relationship dependent on time, allowing for analysis of materials subject to various forces over time.This equation is pivotal for understanding material deformation in engineering and material science, allowing predictions of how materials behave under different stress and strain conditions over time.

To Find:the viscoelastic equation for the four-element model.The Maxwell Viscoelastic Model describes the behavior of viscoelastic materials by combining a spring (elastic) element and a dash pot. (viscous) element in series. The model assumes that the total strain is the sum of the strain in the spring and the strain in the dashpot.The equation for the Maxwell Viscoelastic Model is as follows.Now, let's derive an equation that shows the change in strain over time using the Maxwell Viscoelastic Model. To do this, we'll assume that the stress is constant over time which means the material is subjected to a constant load.Therefore, the equation becomes.Next, isolate the strain rate term to one side of the equation:Now, divide both sides by .Rearrange the equation to highlight the linear relationship.The equation above shows the linear relationship between the strain rate and the total strain (ε(t)) for a material described by the Maxwell Viscoelastic Model when subjected to a constant load (σ). The coefficient represents the slope of the linear relationship and is the y-intercept.

Answered: Derive the viscoelastic equation for…

Engineering

Mechanical Engineering

Derive the viscoelastic equation for the four-element model.

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

See similar textbooks

Related questions

Q: Using the equation Kic= of Vna , plot the fracture stress (MPa) for aluminium alloy 7075- T651…

Q: There are a variety of models that can be used to describe viscoelastic behaviors. In which of the…

A: Viscoelasticity is the property where a material behaves like both liquid as well as solid. When a…

Q: Steady-state creep rate data are given here for some alloy taken at 200°C (473 K): i Es (h-¹) 2.2 x…

Q: Are you please able to include a free body diagram for the question and solution above

A: Given data:- To find:-the state of stress on an element if it is oriented 30° clockwise from the…

Q: You have 4 springs, each being 3.5, 6, 8.5, 11 N/m What's the smallest stiffness you can make using…

Q: When a viscoelastic material is loaded and unloaded, the unloading curve is different to the loading…

A: The stress-strain curve of a viscoelastic material exhibits hysteresis, with the area of the loop…

Q: what is the equation and the units of measure for ductility?

Q: 2) A 1 in Dia x 3 in long mild steel hammer head is to be heat treated for surface hardening. If the…

Q: 2. Find what direct tensile stress load if suddenly applied on a bar 40mm diameter and 3m long would…

Q: A central through-the-thickness crack, 50 mm long, propagates in a thermoset polymer in an unstable…

A: Given data: Applied stress, σ = 5 MPa Length of the crack, a = 50 mm For a centre through the…

Q: Silica (silicon oxide) space shuttle tile is bonded to silicone adhesive. (Look up the CTE for both…

A: Given Temperature, T1 = 298 K T2 = 348 K Coefficient of thermal…

Q: Find the maximum biaxial stress,0x= -20, oy= -30, o > 0, that one can apply to a 60° lamina of…

Q: A material has a yield strength of 1150Mpa and a fracture toughness of 47 Mpa m^1/2. If the applied…

A: Data Given : We need to find the internal crack that can lead to failure.

Q: 4. The mechanical properties of a given titanium alloy (Ti-6Al-4V) are E = 114 GPa, σy = 880 MPa and…

Q: an you represent strain as a function of time using the Maxwell Viscoelastic Model.

A: The viscoelastic material indicates dual characteristics like viscous and elastic behavior under…

Question

**Derive the Viscoelastic Equation for the Four-Element Model**

In viscoelasticity, the four-element model, also known as the Burger’s model, is used to describe the behavior of materials that exhibit both viscous and elastic characteristics when undergoing deformation.

This model consists of two main components connected in series:
1. **Kelvin-Voigt Model**: A spring and dashpot in parallel.
2. **Maxwell Model**: A spring and dashpot in series.

### Derivation Steps:

1. **Model Components**:
- *Maxwell Arm*: Includes a spring (\(E_1\)) and dashpot (\(\eta_1\)).
- *Kelvin-Voigt Arm*: Includes a spring (\(E_2\)) and dashpot (\(\eta_2\)).

2. **Constitutive Equations**:
- For the Maxwell Model:
\[
\sigma(t) = E_1 \cdot \varepsilon_1(t) + \eta_1 \cdot \frac{d\varepsilon_1}{dt}
\]
- For the Kelvin-Voigt Model:
\[
\sigma(t) = E_2 \cdot \varepsilon_2(t') + \eta_2 \cdot \frac{d\varepsilon_2}{dt}
\]

3. **Connection in Series** means:
\[
\varepsilon(t) = \varepsilon_1(t) + \varepsilon_2(t')
\]
where \(\varepsilon(t)\) is the total strain.

4. **Overall Viscoelastic Equation**:
- By combining the contributions from both components, the complete equation is derived to predict the viscoelastic response of the four-element model.

The final form describes the stress-strain relationship dependent on time, allowing for analysis of materials subject to various forces over time.

This equation is pivotal for understanding material deformation in engineering and material science, allowing predictions of how materials behave under different stress and strain conditions over time.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1: determine the viscoelastic equation for the four-element model.

VIEW

Step 2: Find the equation of Maxwell Viscoelastic Model.

VIEW

Step 3: Find the equation for the strain rate.

VIEW

Solution

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 4 steps with 12 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.

Similar questions

(a) An AISI steel plate has width W = 30 cm and a central crack with size of 3 mm. The plate is under a uniform stress. Find the maximum value of the stress is KIc = 50 MPa m1/2. (b) If the part has to operate at a stress of 1,500 MPa, compute the maximum crack size that the plate can have.
Problem#3: You need to design a metallic glass for use in industry. The glass transition temperature of the metallic glass is 250°C. While in use, the sample will be subjected to a permanent tensile stress for tuse=6 years and at a temperature of Tuse=230°C. You need to do the necessary creep experiment in the lab to predict the performance of this material by applying the same stress at different temperature and different time. The creep test will be carried out for tiab=6 hours. At what temperature, Tlab, can this test be performed in the lab under the same tensile stress? [Hint: Use WLF and VFT equations for two cases: 1) Tref=Tg =250 °C and 2) Tr=260 °C.]
6.27 Compute the dislocation density in tungsten if the flow stress is controlled by the stress necessary to operate a Frank--Read source. The shearstress in the slip plane is 50 MPa. Take G = 166 GPa.
Case Study Calculate the Tensile Toughness Tensile toughness measures the total energy absorbed prior to fracture and is different from fracture toughness and impact toughness. Fracture toughness measures the resistance to fracture when a flaw (i.e. crack) is present in the material. Since it is nearly impossible and costly to manufacture materials with zero or near zero defects, fracture toughness is a major consideration for all structural materials. Impact toughness measures a materials ability to withstand an impact blow (i.e. high strain rates) and is particularly important in assessing the ductile-to-brittle transition behavior (more details later). Impact toughness is also commonly referred to as notch toughness. The tensile toughness can be estimated using the stress-strain curve by measuring the area under the curve (both the elastic and plastic regions). Tensile Toughness = f Stress, σ Ty- Ey Efracture ETS Ur Ey Efracture o de = U₁ + SETS o de + Sefracture o de Ey Ey o de =…
Q3: Two material were tested in tension, the first is aluminum alloy has stress-strain curve define by 300 MPa was fracture after 35% reduction in area. The other a copper alloy has stress-strain curve define by=100+120 MPa was fracture after 98% elongation. Which alloy have more toughness? Support your answer with the sketch.?
Compute the ultimate strength for a composite which uses Aramid fibers and Polyester matrix. The volume fraction of fibers is 45 % and matrix is 55 %. Note- use chart in image