I use LGPL licensed software to create some digital assets that contain original designs. [...] The primary use of the assets is the creation of physical objects, via machining or 3D printing. I apply a strong copyleft license, say the GPL, to my copyright-able work.
If end users manufacture and sell such physical objects, do they have to provide the GPL along with the physical objects they go on to create using my digital designs?
TL;DR: If the design in question is a piece of artistic work, then it's almost always yes (but "artwork" is not to be confused with technical patterns for printing, which is also technically called "artwork"). But if the object is only designed for practical use, e.g. in machines, usually no.
In particular, if the physical object is designed for utilitarian purposes, in other words, for practical use (e.g. a machine or its component), and that the object contains absolutely no visible artistic work, it's often not copyrightable. This remains true regardless of its license status, proprietary or copylefted. There are exceptions that require a case-by-case analysis (integrated circuit design is a notable example) but for the most part, these objects are not covered by copyright laws, especially in the US. Other parts of the world have similar laws.
What does the FSF say
This is an answer for GPL, but it applies equally to LGPL and proprietary designs.
Q: Can I use the GPL to license hardware? (#GPLHardware)
Any material that can be copyrighted can be licensed under the GPL. GPLv3 can also be used to license materials covered by other copyright-like laws, such as semiconductor masks. So, as an example, you can release a drawing of a physical object or circuit under the GPL.
In many situations, copyright does not cover making physical hardware from a drawing. In these situations, your license for the drawing simply can't exert any control over making or selling physical hardware, regardless of the license you use. When copyright does cover making hardware, for instance with IC masks, the GPL handles that case in a useful way. 
What do I say
Useful Art Doctrine
First, copyright laws in general covers expression of ideas, but not ideas and facts themselves, this is known as idea-expression dichotomy. U.S. copyright law is 17.102(b) says,
In no case does copyright protection for an original work of authorship extend to any idea, procedure, process, system, method of operation, concept, principle, or discovery, regardless of the form in which it is described, explained, illustrated, or embodied in such work.
This is why we often say that "algorithms are not copyrightable but their implementations can be" in software.
Furthermore, unlike software that exists solely as abstract information and thus extensively covered by copyright laws, physical objects do not follow the same rules. Copyright laws often distinguish a design blueprint and a physical object manufactured from that blueprint. If a blueprint is copyrightable, it does not necessarily follows that the physical objects manufactured using the blueprint are copyrightable too. For example, under US copyright laws, the Useful Articles doctrine  states that:
A “useful article” is an object having an intrinsic utilitarian function that is not merely to portray the appearance of the article or to convey information. Examples are clothing, furniture, machinery, dinnerware, and lighting fixtures. An article that is normally part of a useful article may itself be a useful article, for example, an ornamental wheel cover on a vehicle.
Copyright does not protect the mechanical or utilitarian aspects of such works of craftsmanship. It may, however, protect any pictorial, graphic, or sculptural authorship that can be identified separately from the utilitarian aspects of an object. Thus, a useful article may have both copyrightable and uncopyrightable features.
This is an intentional decision to draw the boundary between copyright laws and patent laws. Copyright mostly covers artistic and creative works, while patents mostly cover practical inventions such as machinery.
Furthermore, it's also worth noting that in addition to the utility patent that we're familiar with, one may apply a special "design patent" to an object's decorative features. However, you cannot game the Useful Articles doctrine by hiding a piece of invisible artwork inside the object, or trying to patent functional aspects the object.
In that case, the court denied a request for a preliminary injunction by the "inventor" of a plastic container for an electric dimmer switch, finding that the container followed precisely the contours of the switch that it enclosed, and was, therefore, "primarily functional" rather than "primarily ornamental." [...] Power Controls, 806 F.2d at 240. Design patents sometimes issue and are then invalidated (for want of ornamentality) on articles that are concealed in use. A vivid example is a septic tank. See C & M Fiberglass Septic Tanks, Inc. v. T & N Fiberglass Mfg. Co., 214 U.S.P.Q. (DNA) 159 (D. S.C. 1981). In such cases the patentee is probably trying to capture some functional attribute. 
In addition to utility patents, the Patent Act "provides limited monopolies on product designs that are new, original, and ornamental. Congress created the design patent to promote products possessing
"grace and pleasing appearance." The integrated circuit design, however, is concealed during its use and therefore has no appearance,pleasing or otherwise.'More significantly, chips failed to meet design patent standards because their design is dictated solely by function and not by cosmetic values." Thus, design patents were also inappropriate for semiconductor 
Implication in Open-Source Hardware Design
The practical implication is that, since copyright laws are not intended to stop someone else from making the same object, in open-source hardware design, licensing terms are much weaker than its use in software.
Hardware in general is not copyrightable  (excluding firmware and its exterior decorative arts or industrial designs).
If the hardware contains original innovations, a utility patent may be applied. If it contains decorative arts or industrial designs, it's copyrighted, and one can also apply a design patent. But if it's just an implementation of a preexisting design, and without artistic decorative features, it cannot be copyrighted nor patented, and there's almost nothing to stop someone else from replicating it.
Case Study - Integrated Circuit Design
An interesting case study is the copyright status of circuit layout in integrated circuits. Before 1984, chip designs could not be effectively copyrighted . Because copyright laws, in general, do not cover the physical chip.
As a result, it was legal to inspect a chip under a microscope and create your own with the same patterns, as long as:
- The chip was legally obtained.
- Original engineering drawings were not copied without permission.
- The circuitry on the chip was not covered by patents (which could be difficult to obtain, since many chips do not invent original circuitry, it's only the same thing on silicon).
- The silicon was not created as a work of art (merely etching artwork does not count, see the discussion of "concealed in use" above).
- The chip does not contain copyrightable software as code (microcode is a borderline case, whether it was software or hardware was heavily debated in many legal cases).
Thus, the semiconductor industry in the 1980s was highly active with many authorized second-source chips and unauthorized clones alike.
The wide range of uses for semiconductors in both new products and redesigned existing products created a huge market for this new type of electronic components. United States manufacturers feared a concerted effort by foreign semiconductor firms, especially the Japanese, to take over the U.S. market. Thus, U.S. manufacturers began to seek protection for their chips in 1979,51 As a first step, industry members tried to obtain protection by fitting their chips into existing categories of intellectual property.
Legislative investigators then turned to the Copyright Act, because its subject matter coverage seemed expansive enough to cover chips. Among the subject matter covered by this statute were pictorial, graphic, and sculptural works, which appeared to include drawings made of a chip's layout. Copyright law also offered a way around difficulties posed by changes in technology, because the Copyright Act explicitly addressed derivative works. This avenue failed, however, because the Copyright Act as applied has distinguished between technical drawings and the objects produced using them. Although a technical drawing itself can be protected, the Copyright Act does not protect the objects made using the drawing. Thus the Act did not address the semiconductor manufacturer's main concern: prohibiting others from copying its circuit layout and selling the results.
A second difficulty in using the Copyright Act to cover chips was the "useful article doctrine."65 Under this doctrine, the Copyright Act does not give protection to utilitarian objects unless their useful functions can be separated physically or conceptually from their aesthetic aspects. For example, a belt buckle that appears as a free-form metal sculpture is conceptually separable from its useful function. A mask work, however, functions only as part of the manufacturing process and has no real aesthetic aspects, making it a purely useful article. The Copyright Office has consistently denied protection to useful articles, and has refused to accept chips as copies of masks under the Copyright Act.
After determining that existing intellectual property statutes could not appropriately be applied to mask works, Congress decided that the correct answer to the problem lay in a new sui generis form of protection. Although the new act was modeled after the Copyright Act, it created a new form of industrial intellectual property.
Due to the high cost and strategic importance of the semiconductors industry, the non-copyrightable status of integrated circuit received special government attention after extensive lobbying from the industry. Thus, a special form of copyright was created just for integrated circuit layouts, named Semiconductor Chip Protection Act of 1984. It has a term of 10 years. In 1989 similar laws were included in international trade agreements.
The same argument on the non-copyrightable status of chips also applies to other kinds of practical physical objects, such as machine parts. In particular, the argument on the non-copyrightable status of integrated circuit design can be applied to circuit board design almost in verbatim. So whenever you see a copyright symbol on a circuit board, it's probably nonsense, at least without firmware or software).
 Semiconductor Chip Protection: Changing Roles for Copyright and Competition, James Chesser, Virginia Law Review, Vol. 71, No. 2 (Mar., 1985), pp. 249-295, doi: 10.2307/1073018