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Linear Viscoelastic Model
Viscoelastic Materials
Viscoelastic Materials consist of polymers of variable chain length and filters resulting in the following stress strain diagram.
Figure 12.1: Stress strain diagram for viscoeleastic material
Slope of the major axis of the ellipse is a measure of material stiffness while the ratio of the minor to major axis is a measure of damping.
Considering shear deformation of the viscoelastic materials the stress-strain relationship may be written as
where 
=shear stress, φ(t) is the shear strain. G' is the real part of the shear modulus (Storage Modulus) and G" is the imaginary part of the shear modulus (Loss Modulus).
Stress-strain relationship
The generalised stress-strain relationship in a polymer could be written as
(12.2)
where σ is the stress and ∈ is the strain. In this relationship, if all the coefficients a0,......an and b0,......bm are constants, the material is referred to as linear viscoelastic. The reason the material is called thus is that we can, combining linear springs (elastic) and dashpots (viscous), arrive at the stressstrain relationship with constant coefficients.
Kelvin-Voight Model
Let us consider the force-deformation relationship of a two parameter model in the parallel combination of a linear spring of stiffness and a viscous dashpot of damping coefficient n (Fig.12.1).
Fig12.1: Kelvin-Voight model
The stress-strain relationship of the spring and damper elements could be written as
where, the subscript "s" and "d" stand for spring and damper respectively.
Since, for the case the overall strain ∈ is such that 
hence,
Maxwell Model
Let us now consider the Maxwell model of viscoelastic materials. The model is described in the Fig 12.2
Figuer 12.2: Maxwell model
In this model, you may note that the spring and dashpot are connected in series such that
Since in this case
Differentiating the above equation, we get
Hence, for the model
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FAQs on Linear Viscoelastic Model - Civil Engineering (CE)
| 1. What is a linear viscoelastic model? |  |
A linear viscoelastic model is a mathematical model used to describe the behavior of materials that exhibit both elastic and viscous properties under applied stress or strain. It assumes that the material response is linear, meaning the stress or strain is directly proportional to the applied force or deformation. This model is commonly used in various industries, including polymer science, rheology, and material engineering.
| 2. What are the key features of a linear viscoelastic model? | |
The key features of a linear viscoelastic model include:
- Elastic behavior: The material can store and release energy when subjected to stress or strain, exhibiting a reversible deformation.
- Viscous behavior: The material dissipates energy as heat during deformation, resulting in irreversible deformation.
- Linear response: The stress or strain is directly proportional to the applied force or deformation within the material's linear range.
- Time-dependent behavior: The material response depends on the rate and duration of the applied stress or strain.
| 3. How is a linear viscoelastic model different from a linear elastic model? | |
A linear viscoelastic model differs from a linear elastic model in that it considers both elastic and viscous behavior of materials. In a linear elastic model, the material only exhibits elastic behavior, meaning it can store and release energy without dissipating any. On the other hand, a linear viscoelastic model incorporates viscous behavior, where the material dissipates energy as heat during deformation. This allows the linear viscoelastic model to describe materials that exhibit time-dependent responses, such as polymers and biological tissues.
| 4. What are some applications of linear viscoelastic models? | |
Linear viscoelastic models have various applications in different fields, including:
- Polymer science: They are used to study the behavior of polymer materials under different conditions, such as temperature, frequency, and strain rate.
- Rheology: They help analyze the flow and deformation properties of complex fluids, such as paints, adhesives, and food products.
- Material engineering: They aid in predicting and understanding the mechanical behavior of materials used in structural applications, such as metals, composites, and elastomers.
- Biomechanics: They are utilized to study the viscoelastic properties of biological tissues, such as bones, muscles, and cartilage, to better understand their mechanical response and design biomedical devices.
| 5. How are linear viscoelastic models experimentally characterized? | |
Linear viscoelastic models can be experimentally characterized using techniques such as dynamic mechanical analysis (DMA) or rheometry. These techniques involve subjecting the material to oscillatory stress or strain under controlled conditions and measuring the resulting mechanical response. The data obtained from these experiments, such as storage modulus, loss modulus, and phase angle, can then be used to fit the material's behavior to a specific linear viscoelastic model, allowing for further analysis and prediction of its viscoelastic properties.
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