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Given the Cauchy stress tensor at a point in an arbitrary body, find the following: a) The traction vector T acting on a surface that passes through the point and is parallel to the plane 2x1+2x2+3x3=0 (units: Mpa). b) The traction vector T acting on a surface normal to the vector [4, -2, 1].

Consider a bar with a uniform rectangular cross section and area A. The bar is loaded uniaxially with a compressive force F along the xy-axis as shown below. In addition, ambient pressure P acts normal to all external surfaces of the bar. Suppose a plane is cut at a 40 degree angle as shown in the figure. Write the normal vector, n, that defines the 40 degree plane. Note that the plane is not tilted in the x direction. b) Write the Cauchy stress tensor for this case of uniaxial loading plus ambient pressure. Use the Cauchy stress tensor and the normal vector to calculate the traction vector, T, on the plane defined by the normal vector, n. Find the resultant normal stress and shear stress from T.

Explain the key differences in the mechanical behavior of ligaments versus tendons, and the underlying structural reasons. Plot a stress-strain graph for tendon and ligament on the same axes (labeled clearly with units) to illustrate your points.

Suppose you are testing the permeability of a cartilage sample. You apply a known pressure gradient across the sample with a permeability k and dimensions as shown below on left, and measure flow Q in the xy direction. You then compress the sample in the x2 direction to the dimensions shown below on right, and determine Q has been reduced by 80 percent. a) How much has the permeability changed after compression? Express your answer as a percentage of the original permeablity k. Does your answer make sense? Why or why not? The graph on the left shows the displacement of a piece of cartilage under compression by a constant load. b) What type of behavior is the cartilage exhibiting? c) Remembering what we know about the composition of cartilage, what is happening during phase A? B? C?

Given a cube in equilibrium under the following uniform loading conditions. All external surfaces of teh cube have a surface area of 1 m^2. All forces labeled with an F (blue) act normal to a surface, while all forces labeled with a V (red and purple) are shear forces. All forces have units=N. a) Write the Cauchy Stress Tensor for this loading condition. b) Find the prinicpal stress components S1, S2, and S3. c) For each principal stress, find the normal vector n that defines the plane on which the principal stress acts.

You are developing a biomaterial to replace damaged ligaments. Your biomaterial performs well over short times, but the relaxation time of the biomaterial is worrisome. If the force in the material is greater than 0.005 N for longer than 1 minute after an applied step in deformation, the biomaterial begins to break down. You know that the maximum deformation the biomaterial will be exposed to is 0.1 m and that any springs in the model will have k=3 N/m and any dashpots will have h=50 Ns/m. a) You are debating between the Maxwell and Voigt models. Which should you use? Why? b) Will the biomaterial be able to withstand the maximum deformation without degrading? Show some math to back up your answer. c) How would you change k to improve perrformance? What about h? show some math to back up your answer. Plot the relaxation function of your model in response to the maximum deformation. Include the suggestions for improvement you made in part c.

Consider a bar with a uniform rectangular cross section and area A. The bar is loaded uniaxially with a compressive force F along the xy-axis as shown below. In addition, ambient pressure P acts normal to all external surfaces of the bar. Suppose a plane is cut at a 40 degree angle as shown in the figure. Write the normal vector, n, that defines the 40 degree plane. Note that the plane is not tilted in the x direction. b) Write the Cauchy stress tensor for this case of uniaxial loading plus ambient pressure. Use the Cauchy stress tensor and the normal vector to calculate the traction vector, T, on the plane defined by the normal vector, n. Find the resultant normal stress and shear stress from T.

Explain the key differences in the mechanical behavior of ligaments versus tendons, and the underlying structural reasons. Plot a stress-strain graph for tendon and ligament on the same axes (labeled clearly with units) to illustrate your points.

Suppose you are testing the permeability of a cartilage sample. You apply a known pressure gradient across the sample with a permeability k and dimensions as shown below on left, and measure flow Q in the xy direction. You then compress the sample in the x2 direction to the dimensions shown below on right, and determine Q has been reduced by 80 percent. a) How much has the permeability changed after compression? Express your answer as a percentage of the original permeablity k. Does your answer make sense? Why or why not? The graph on the left shows the displacement of a piece of cartilage under compression by a constant load. b) What type of behavior is the cartilage exhibiting? c) Remembering what we know about the composition of cartilage, what is happening during phase A? B? C?

Given a cube in equilibrium under the following uniform loading conditions. All external surfaces of teh cube have a surface area of 1 m^2. All forces labeled with an F (blue) act normal to a surface, while all forces labeled with a V (red and purple) are shear forces. All forces have units=N. a) Write the Cauchy Stress Tensor for this loading condition. b) Find the prinicpal stress components S1, S2, and S3. c) For each principal stress, find the normal vector n that defines the plane on which the principal stress acts.

You are developing a biomaterial to replace damaged ligaments. Your biomaterial performs well over short times, but the relaxation time of the biomaterial is worrisome. If the force in the material is greater than 0.005 N for longer than 1 minute after an applied step in deformation, the biomaterial begins to break down. You know that the maximum deformation the biomaterial will be exposed to is 0.1 m and that any springs in the model will have k=3 N/m and any dashpots will have h=50 Ns/m. a) You are debating between the Maxwell and Voigt models. Which should you use? Why? b) Will the biomaterial be able to withstand the maximum deformation without degrading? Show some math to back up your answer. c) How would you change k to improve perrformance? What about h? show some math to back up your answer. Plot the relaxation function of your model in response to the maximum deformation. Include the suggestions for improvement you made in part c.

Two viscoelastic biological tissues of unknown origin and identity are tested to determine their mechanical properties. For specimen A, the force was measured in reponse to a step deformation. for specimen B, the deformation was measured in response to a step force. The results (red line) are shown below. a) Design a viscoelastic model to represnt Specimen A. Specify the name of the model chosen, and the elasticity or viscosity values for each component. b) Draw the expected creep response of specimen A to a step force of 1 kN. State the equation of the expected response. c) Design a viscoelastic model to represent specimen B. Specify the name of the model chosen, and the elasticity or viscosity values for each component. d) Draw the expected force reponse of specimen B to a step deformation of 1 cm. State the equation of the expected response.

Kim is driving down the road in her Camaro Z28 when David pulls up next to her in his Honda Civic and challenges her to a burn out competition. Kim knows you are taking biomechanics and asks if she should accept the challenge. Kim gives you the following numbers.Honda Civic: 188 Nm torque, 2886 lb, 24 inch tire diameter, 60 percent of weight over drive tires. Camaro Z28: 441 Nm torque, 3450 lb, 25 inch tire diameter, 40 percent of weight over drive tires. Kim also notes that Davit put Formula racing tires on his Honda to increase his stree cred while she has hi performance stree tires. Kim explains that a car's torque rating is equivalent to a moment applied at the center of the wheel. The frictional force between the tire and the road produces a moment that opposes the car's torque. If the torque is greater than the frictional force, the car's tires will spin in place. Using the diagram, calculate the moment produced by the frictional force for each car. Should Kim accept the challenge?

Scientists at a medical device company are testing a new bone graft design. They want to determine which material to make the bone graft from. They have developed two tests to evaluate the new design.

A 1 meter beam is firmly attached to a wall on one end. The other end of the beam is supported by a roller, generating a reaction force of 15 N acting in the positive y direction. the wight of the beam, 10 N, acts at 0.5 m. A 40 N force is exerted at 0.25 m, and a cable connection at 60 degrees acts at 0.75 m. a) Draw a FBD of the beam. b) Calculate the values of the reaction forces and moment generated at the fixed support. c) Calculate the internal forces and moment as a function of distance x along the beam. Plot the functions. You must use the coordinate system provided, where the fixed support connection corresponds to x=0.0.

Consider the position of the head and the neck shown below. Also shown are the forces acting on the head. The head weights W=50 N and its center of gravity is located at C. Fu is the magnitude of the resultant force exerted by the neck extensor muscles, which is applied on the skull at A. The atlantooccipital joint is located at B. For this flexed position of the head, the line of action of the neck muscle force makes an angle of 30 degrees and the line of action of the joint reaction force makes an angle of 60 degrees with the horizontal. Draw a free body diagram for the head. What tension must the neck extensor muscles exert to support the head? What is the force applied on the first cervical vertebra at the atlantooccipital joint?

A sample of cortical bone is extracted from a tibia as shown below. The packed cylinders show how the osteons are aligned. Treat the bone sample as an orthotropic hookean elastic solid. Use the coordinate system we developed in class wherein we assigned numerical values the the radial, circumferential, and longitudinal directions of the long bone. Based on the given material constants for the 3 loading conditions shown below, calculate the small strain tensor for each loading condition.

Kim is driving down the road in her Camaro Z28 when David pulls up next to her in his Honda Civic and challenges her to a burn out competition. Kim knows you are taking biomechanics and asks if she should accept the challenge. Kim gives you the following numbers.Honda Civic: 188 Nm torque, 2886 lb, 24 inch tire diameter, 60 percent of weight over drive tires. Camaro Z28: 441 Nm torque, 3450 lb, 25 inch tire diameter, 40 percent of weight over drive tires. Kim also notes that Davit put Formula racing tires on his Honda to increase his stree cred while she has hi performance stree tires. Kim explains that a car's torque rating is equivalent to a moment applied at the center of the wheel. The frictional force between the tire and the road produces a moment that opposes the car's torque. If the torque is greater than the frictional force, the car's tires will spin in place. Using the diagram, calculate the moment produced by the frictional force for each car. Should Kim accept the challenge?

Scientists at a medical device company are testing a new bone graft design. They want to determine which material to make the bone graft from. They have developed two tests to evaluate the new design.

A 1 meter beam is firmly attached to a wall on one end. The other end of the beam is supported by a roller, generating a reaction force of 15 N acting in the positive y direction. the wight of the beam, 10 N, acts at 0.5 m. A 40 N force is exerted at 0.25 m, and a cable connection at 60 degrees acts at 0.75 m. a) Draw a FBD of the beam. b) Calculate the values of the reaction forces and moment generated at the fixed support. c) Calculate the internal forces and moment as a function of distance x along the beam. Plot the functions. You must use the coordinate system provided, where the fixed support connection corresponds to x=0.0.

Consider the position of the head and the neck shown below. Also shown are the forces acting on the head. The head weights W=50 N and its center of gravity is located at C. Fu is the magnitude of the resultant force exerted by the neck extensor muscles, which is applied on the skull at A. The atlantooccipital joint is located at B. For this flexed position of the head, the line of action of the neck muscle force makes an angle of 30 degrees and the line of action of the joint reaction force makes an angle of 60 degrees with the horizontal. Draw a free body diagram for the head. What tension must the neck extensor muscles exert to support the head? What is the force applied on the first cervical vertebra at the atlantooccipital joint?

A sample of cortical bone is extracted from a tibia as shown below. The packed cylinders show how the osteons are aligned. Treat the bone sample as an orthotropic hookean elastic solid. Use the coordinate system we developed in class wherein we assigned numerical values the the radial, circumferential, and longitudinal directions of the long bone. Based on the given material constants for the 3 loading conditions shown below, calculate the small strain tensor for each loading condition.

You perform a pipet creep test on a spherical cell as shown below on the left. a) Find the tension in the cell membrane when Pp=-75 mN/um^2. b) You know that this cell's behavior is best modled as a Voight fluid with a cortical tension of 45 mN/um, as shown below on the right. A step pressure Pp=-75 mN/um^2 is applied in a creep test. Based on the graphs shown below, solve for the viscosity and elasticity.

For the RBC micropipet experiment below, a micropipet pressure Pp of -20 dynes/cm^2 is applied. The pipet radius Rp is 1.2 um, the maximum cell radius Rc is 4.3 um, Lp is 0.9 um, and P0 is 0 dynes/cm^2. Find the membrane tension component T at the interface of the micropipet and cell. b) We solved for the membrane tension component T in part a, Now, assume that T is the same at the bottom of the cell and that the cell can be represented as a simple hockey puck disc shape. A free body diagram of the bottom of the disk is shown below, where Rc is the cell radius (4.3 um). h is the membrane thickness, and P0 is the external pressure. Perform a force balance to generate an equation for Pi and calculate Pi. c) Like most biological material, RBC exhibit characteristics of both solid and fluid materials. During aspiration, RBC membranes can resist aspiration due to their shear elasticity when Lp is greater than Rp.

Consider the bar in equilibrium shown below. It is subjected to four loads. The first is a shear force of 2000 N that acts on the top and bottom surfaces. Note that reaction forces to maintain rotational equilibrium in response to V exist but are not shown. The second is a uniform hydrostatic pressure P of 10 kPa. The third is a moment M of -1000 Nm which acts uniformly throughout the bar and causes a twist about the x1 axis. Note that for the moment M, the neutral axis of bending is defined by the x3 axis. The fourth is a non uniform normal force F of 10000X3 N that acts on the top and bottom surfaces in the direction drawn. The length of the bar in the x3 direction is 1 m, the height of the bar in the x2 direction is 0.3 m, and the width of the bar in the x1 direction is 0.2 m. a) Write the Cauchy stess tensor for this loading scenario. b) Write the unit normal vector, n, that defines the 35 degree plane shown in the figure. Note that if you are looking into the bar along the x1 axis, the plane is not visible. c) Use the Cauchy stress tensor and the normal vector to calculate the traction vector T on the plane at the lower surface of the bar. d) Find the resultant normal stress and shear stress from T.

In runners, an overuse injury created by overpronation of the foot can lead to "compartment syndrome". This syndrome is characterized by high pressure and swelling in the soleus muscle behind the tibia. You want to estimate how much wall stress is created in the connective tissue sheath surrounding the muuscle when pressure in the muscle is elevated. You model the muscle sheath as a cylinder as shown to the right. Pressure inside the cylinder is 10 mmHg. The wall of the cylinder has a uniform thickness of 0.2 cm. The diameter of the cylinder is 4 cm. Calculate the mean normal stress in the x direction in the wall of the cylinder.

A patient with hypertension has developed a saccular aneurism in the brain. The circumferential wall stress in the aneurism can be modeled by the equation sigma=pa^2/(2ah+h^2). If the wall stress exceeds 7*10^5 dynes/cm^2, the aneurism will rupture, causing a stroke. The patient's average blood pressure is 160 mmHg. The inner radius of the aneurism is 1.8 mm and the wall thickness is 0.3 mm. a) What is the cirumferential wall stress in the aneurism? b) Based on the equation for wall stress and what we have talked about in class, what might the body attempt to do to reduce the wall stress in the aneurism? c) The patient is given a vasodilator to reduce their blood pressure. Their blood pressure is reduced to 140 mmHg, but the vessel wall in the aneurism relaxes and the inner radius increases to 3.0 mm. What is the new value of the wall stress? Was the medication helpful in preventing aneurism rupture?

A complex viscoelastic material behaves as shown below. The material exhibits creep, relaxation and recovery. a) What aspects of this behavior would the Maxwell model fail to predict? What aspects of this behavior would the Voigt model fail to predict? b) A more complicated model, the Kelvin model, that correctly predicts creep, relaxation and recovery is shown below. Using the information provided, plot F(t) and u(t) for 0 to t to 150 seconds. c) Imagine the material you are modeling is bone. When exposed to large constant forces, bone may exhibit creep, a gradual increase in deformation. In terms of the model, what value(s) could we change to reduce this behavior without reducing the applied force F0? d) Now imagine that the material you are modeling is a tendon. When a constant deformation is applied, the tendon experiences relaxation, a gradual decrease in the internal force. In terms of the model, what value(s) could we change to reduce this behavior?

The pressure drop along a capillary is ordinarily about 12 mmHg. Suppose a white blood cell (WBC) has become lodged in a capillary as shown in the figure. Assume venous pressure Pv remains constant, the coefficient of static friction between the cell and the capillary wall is 0.2, the normal stress exerted by the deformed cell on the capillary wall is 12 mmHg, L=10 um, and D=4.5 um. a) How much higher will arterial pressure Pa ahve to rise before the WBC cell begins to move? b) Based on our discussion of WBC viscoelasticity in class, what would you predict the apparent viscosity of the cell to be if it deformed into this shape in only 1 second or less? Briefly justify your answer. c) Same as part b, except assume the time of WBC deformation is 30 seconds. d) Suppose that the WBC is dislodged from the capillary and isn ow freely circulating through an artery whose diameter is bigger than the diameter of the cell. Assuming that the WBC is spherical and incompressible, what are the radius and the aparent change in surface area of the cell?

For the RBC micropipet experiment below, a micropipet pressure Pp of -20 dynes/cm^2 is applied. The pipet radius Rp is 1.2 um, the maximum cell radius Rc is 4.3 um, Lp is 0.9 um, and P0 is 0 dynes/cm^2. Find the membrane tension component T at the interface of the micropipet and cell. b) We solved for the membrane tension component T in part a, Now, assume that T is the same at the bottom of the cell and that the cell can be represented as a simple hockey puck disc shape. A free body diagram of the bottom of the disk is shown below, where Rc is the cell radius (4.3 um). h is the membrane thickness, and P0 is the external pressure. Perform a force balance to generate an equation for Pi and calculate Pi. c) Like most biological material, RBC exhibit characteristics of both solid and fluid materials. During aspiration, RBC membranes can resist aspiration due to their shear elasticity when Lp is greater than Rp.

Consider the bar in equilibrium shown below. It is subjected to four loads. The first is a shear force of 2000 N that acts on the top and bottom surfaces. Note that reaction forces to maintain rotational equilibrium in response to V exist but are not shown. The second is a uniform hydrostatic pressure P of 10 kPa. The third is a moment M of -1000 Nm which acts uniformly throughout the bar and causes a twist about the x1 axis. Note that for the moment M, the neutral axis of bending is defined by the x3 axis. The fourth is a non uniform normal force F of 10000X3 N that acts on the top and bottom surfaces in the direction drawn. The length of the bar in the x3 direction is 1 m, the height of the bar in the x2 direction is 0.3 m, and the width of the bar in the x1 direction is 0.2 m. a) Write the Cauchy stess tensor for this loading scenario. b) Write the unit normal vector, n, that defines the 35 degree plane shown in the figure. Note that if you are looking into the bar along the x1 axis, the plane is not visible. c) Use the Cauchy stress tensor and the normal vector to calculate the traction vector T on the plane at the lower surface of the bar. d) Find the resultant normal stress and shear stress from T.

In runners, an overuse injury created by overpronation of the foot can lead to "compartment syndrome". This syndrome is characterized by high pressure and swelling in the soleus muscle behind the tibia. You want to estimate how much wall stress is created in the connective tissue sheath surrounding the muuscle when pressure in the muscle is elevated. You model the muscle sheath as a cylinder as shown to the right. Pressure inside the cylinder is 10 mmHg. The wall of the cylinder has a uniform thickness of 0.2 cm. The diameter of the cylinder is 4 cm. Calculate the mean normal stress in the x direction in the wall of the cylinder.

A patient with hypertension has developed a saccular aneurism in the brain. The circumferential wall stress in the aneurism can be modeled by the equation sigma=pa^2/(2ah+h^2). If the wall stress exceeds 7*10^5 dynes/cm^2, the aneurism will rupture, causing a stroke. The patient's average blood pressure is 160 mmHg. The inner radius of the aneurism is 1.8 mm and the wall thickness is 0.3 mm. a) What is the cirumferential wall stress in the aneurism? b) Based on the equation for wall stress and what we have talked about in class, what might the body attempt to do to reduce the wall stress in the aneurism? c) The patient is given a vasodilator to reduce their blood pressure. Their blood pressure is reduced to 140 mmHg, but the vessel wall in the aneurism relaxes and the inner radius increases to 3.0 mm. What is the new value of the wall stress? Was the medication helpful in preventing aneurism rupture?

A complex viscoelastic material behaves as shown below. The material exhibits creep, relaxation and recovery. a) What aspects of this behavior would the Maxwell model fail to predict? What aspects of this behavior would the Voigt model fail to predict? b) A more complicated model, the Kelvin model, that correctly predicts creep, relaxation and recovery is shown below. Using the information provided, plot F(t) and u(t) for 0 to t to 150 seconds. c) Imagine the material you are modeling is bone. When exposed to large constant forces, bone may exhibit creep, a gradual increase in deformation. In terms of the model, what value(s) could we change to reduce this behavior without reducing the applied force F0? d) Now imagine that the material you are modeling is a tendon. When a constant deformation is applied, the tendon experiences relaxation, a gradual decrease in the internal force. In terms of the model, what value(s) could we change to reduce this behavior?

The pressure drop along a capillary is ordinarily about 12 mmHg. Suppose a white blood cell (WBC) has become lodged in a capillary as shown in the figure. Assume venous pressure Pv remains constant, the coefficient of static friction between the cell and the capillary wall is 0.2, the normal stress exerted by the deformed cell on the capillary wall is 12 mmHg, L=10 um, and D=4.5 um. a) How much higher will arterial pressure Pa ahve to rise before the WBC cell begins to move? b) Based on our discussion of WBC viscoelasticity in class, what would you predict the apparent viscosity of the cell to be if it deformed into this shape in only 1 second or less? Briefly justify your answer. c) Same as part b, except assume the time of WBC deformation is 30 seconds. d) Suppose that the WBC is dislodged from the capillary and isn ow freely circulating through an artery whose diameter is bigger than the diameter of the cell. Assuming that the WBC is spherical and incompressible, what are the radius and the aparent change in surface area of the cell?

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