Three-Dimensional Solids and Material Modeling
Comprehensive university textbook unit covering fundamental principles, governing relationships, and engineering applications of Three-Dimensional Solids and Material Modeling.
§5.1 Three-dimensional solid elements
**Target competency.** Derive the appropriate approximate relation, apply it with coherent units, and identify one situation in which it would produce a misleading engineering result. **Primary relation:** `K_e=∫ Bᵀ D B dV`. **Field study:** Model a ribbed housing under a mounting force.
§5.2 Constitutive matrices and Poisson effects
**Target competency.** Derive the appropriate approximate relation, apply it with coherent units, and identify one situation in which it would produce a misleading engineering result. **Primary relation:** `σ = λ tr(ε)I + 2Gε`. **Field study:** Compare axial extension with lateral contraction in a test coupon.
§5.3 Contact, interfaces and load introduction
**Target competency.** Derive the appropriate approximate relation, apply it with coherent units, and identify one situation in which it would produce a misleading engineering result. **Primary relation:** `g_n≥0,\ p_n≥0,\ g_n p_n=0`. **Field study:** Compare a bolt-hole bearing model with an ideal fixed ring.
§5.4 Thermal stress and eigenstrain
**Target competency.** Derive the appropriate approximate relation, apply it with coherent units, and identify one situation in which it would produce a misleading engineering result. **Primary relation:** `ε_mech=ε_total-αΔT I`. **Field study:** Analyze a restrained bar heated by a controlled temperature rise.
§5.5 Material nonlinearity and plastic zones
**Target competency.** Derive the appropriate approximate relation, apply it with coherent units, and identify one situation in which it would produce a misleading engineering result. **Primary relation:** `f(σ,κ)≤0`. **Field study:** Discuss a notched bracket pushed slightly beyond first yield.
Step-by-Step Finite Element Analysis and Computational Mechanics Solutions
Three original questions connect calculations, definitions and interpretation. Open each solution to follow the reasoning.
Problem Statement:
Worked Engineering Derivation:
A small hand-checkable model associated with this unit uses **thermal free expansion**, based on `delta=alpha*DeltaT*L`. Illustrative inputs are **Thermal coefficient=23 µstrain/K, Temperature rise=65 K, Initial length=1.2 m**. Converting the displayed units into SI inside the calculation gives **1.794 mm**. This number comes from the stated elementary relation and is *not* a full FE simulation or a measured response. Recreate the substitution independently, confirm the dimensional units, and explain the physical meaning of the output sign and magnitude. The associated browser model provides bounded parameter changes and a sensitivity plot. For negative outcomes such as compressive stress or reverse heat flow, retain the sign in the balance even if a subsequent design comparison uses absolute magnitude. Compute a second scenario by changing the dominant parameter by ten percent while holding all others fixed. If the governing relation is nonlinear in that parameter, do not assume the output changes by ten percent. For example, beam deflection scales with length cubed under Euler–Bernoulli assumptions; this makes a modest length error important. An area or thickness variation can also change local stress and global stiffness differently. When the relevant approximation instead assumes constant geometry, explain why replacing the original problem by a new geometry requires re-analysis rather than simply scaling the final result.
Problem Statement:
Worked Engineering Derivation:
A small hand-checkable model associated with this unit uses **thermal free expansion**, based on `delta=alpha*DeltaT*L`. Illustrative inputs are **Thermal coefficient=23 µstrain/K, Temperature rise=65 K, Initial length=1.2 m**. Converting the displayed units into SI inside the calculation gives **1.794 mm**. This number comes from the stated elementary relation and is *not* a full FE simulation or a measured response. Recreate the substitution independently, confirm the dimensional units, and explain the physical meaning of the output sign and magnitude. The associated browser model provides bounded parameter changes and a sensitivity plot. For negative outcomes such as compressive stress or reverse heat flow, retain the sign in the balance even if a subsequent design comparison uses absolute magnitude. Compute a second scenario by changing the dominant parameter by ten percent while holding all others fixed. If the governing relation is nonlinear in that parameter, do not assume the output changes by ten percent. For example, beam deflection scales with length cubed under Euler–Bernoulli assumptions; this makes a modest length error important. An area or thickness variation can also change local stress and global stiffness differently. When the relevant approximation instead assumes constant geometry, explain why replacing the original problem by a new geometry requires re-analysis rather than simply scaling the final result.
Problem Statement:
Worked Engineering Derivation:
A small hand-checkable model associated with this unit uses **thermal free expansion**, based on `delta=alpha*DeltaT*L`. Illustrative inputs are **Thermal coefficient=23 µstrain/K, Temperature rise=65 K, Initial length=1.2 m**. Converting the displayed units into SI inside the calculation gives **1.794 mm**. This number comes from the stated elementary relation and is *not* a full FE simulation or a measured response. Recreate the substitution independently, confirm the dimensional units, and explain the physical meaning of the output sign and magnitude. The associated browser model provides bounded parameter changes and a sensitivity plot. For negative outcomes such as compressive stress or reverse heat flow, retain the sign in the balance even if a subsequent design comparison uses absolute magnitude. Compute a second scenario by changing the dominant parameter by ten percent while holding all others fixed. If the governing relation is nonlinear in that parameter, do not assume the output changes by ten percent. For example, beam deflection scales with length cubed under Euler–Bernoulli assumptions; this makes a modest length error important. An area or thickness variation can also change local stress and global stiffness differently. When the relevant approximation instead assumes constant geometry, explain why replacing the original problem by a new geometry requires re-analysis rather than simply scaling the final result.
Problem Statement:
Worked Engineering Derivation:
A small hand-checkable model associated with this unit uses **bending outer-fiber stress**, based on `sigma=M*c/I`. Illustrative inputs are **Bending moment=600 N·m, Outer fiber distance=30 mm, Second moment=800000.0 mm⁴**. Converting the displayed units into SI inside the calculation gives **22.5 MPa**. This number comes from the stated elementary relation and is *not* a full FE simulation or a measured response. Recreate the substitution independently, confirm the dimensional units, and explain the physical meaning of the output sign and magnitude. The associated browser model provides bounded parameter changes and a sensitivity plot. For negative outcomes such as compressive stress or reverse heat flow, retain the sign in the balance even if a subsequent design comparison uses absolute magnitude. Compute a second scenario by changing the dominant parameter by ten percent while holding all others fixed. If the governing relation is nonlinear in that parameter, do not assume the output changes by ten percent. For example, beam deflection scales with length cubed under Euler–Bernoulli assumptions; this makes a modest length error important. An area or thickness variation can also change local stress and global stiffness differently. When the relevant approximation instead assumes constant geometry, explain why replacing the original problem by a new geometry requires re-analysis rather than simply scaling the final result.
Problem Statement:
Worked Engineering Derivation:
A small hand-checkable model associated with this unit uses **bending outer-fiber stress**, based on `sigma=M*c/I`. Illustrative inputs are **Bending moment=600 N·m, Outer fiber distance=30 mm, Second moment=800000.0 mm⁴**. Converting the displayed units into SI inside the calculation gives **22.5 MPa**. This number comes from the stated elementary relation and is *not* a full FE simulation or a measured response. Recreate the substitution independently, confirm the dimensional units, and explain the physical meaning of the output sign and magnitude. The associated browser model provides bounded parameter changes and a sensitivity plot. For negative outcomes such as compressive stress or reverse heat flow, retain the sign in the balance even if a subsequent design comparison uses absolute magnitude. Compute a second scenario by changing the dominant parameter by ten percent while holding all others fixed. If the governing relation is nonlinear in that parameter, do not assume the output changes by ten percent. For example, beam deflection scales with length cubed under Euler–Bernoulli assumptions; this makes a modest length error important. An area or thickness variation can also change local stress and global stiffness differently. When the relevant approximation instead assumes constant geometry, explain why replacing the original problem by a new geometry requires re-analysis rather than simply scaling the final result.